Do the maths: 7 reasons why the obsession with maths doesn’t add up

I'd much rather employ this smart, innovative, humorous and creative person than someone who actually knows how to apply Pythagoras's Theorem. Maths has never been the sole touchstone for being 'smart' or 'employable'. So I'm annoyed that maths has become a weapon of mass distraction in education, a topic full of bluster and exaggeration. Maths is hard to learn, hard to teach and easy to test, in other words the ideal recipe for mass failure. Everyone seems to agree that we have a huge problem in maths. The problem, it seems to me, is a lack of a basic understanding of maths by politicians, employers, even experts in education. So here’s some FAILS, or failures to do the simple maths.

FAIL 1 Numeracy not maths

There is persistent exaggeration in the size of the so-called maths problem. This is caused by people shooting arrows, drawing a chalk circle around the arrow and calling it a bullseye. Typical is the recently formed National Numeracy charity, which claims we have 17 million (nearly half the working population) with poor numeracy. They do this by relying on one ‘survey’ and conflating numeracy with maths, as their definitions are based on GCSE achievement.

This exaggeration is endemic and a simple failure in statistics. The actual needs in the real world match what we call ‘functional maths’: basic numeracy, use of a calculator, some understanding of statistics etc. This is not congruent with what is actually taught in GCSE maths. If they were represented as sets there would be a small overlap. When employers talk about poor maths, they are largely talking about poor numeracy. These are two different things. In fact, almost invariably people conflate and confuse maths with numeracy (or functional maths). A simple Venn diagram is all that’s needed to make this clear.

FAIL 2 Most maths quantifiably irrelevant

What’s the quadratic equation? What’s a surd? When was the last time you divided two fractions? When did you last use algebra? The recent report into the teaching of maths confirmed, yet again, that the curriculum is largely irrelevant to most students, as they are unlikely to use much of it in later life. They rightly recommend a new qualification in functional maths. If GCSE maths were a pie chart, most of it will not be used by the vast majority of people in later life. In any case, if we do need the more complex stuff, we can learn it later. Do the maths. It doesn’t add up.

FAIL 3 Maths is easy to test

Rather than test what really matters in problem solving and real life, we’ve stuck to a lazy and often irrelevant method of testing that puts maths at the top of the tree. Why because it’s easy to test. Maths problems have single solutions and are therefore easy to test. Nevertheless, problems, largely of calculation, are perceived as being a good test of one’s ability in a general sense. This is nonsense. Maths problems are rarely realistic. Nobody goes around using maths to share marbles, split up pizzas, share out cakes at parties or dilute orange juice. There is a critical failure to ‘bridge’ between the real world and its representation in mathematical language. But in an age of perpetual testing, maths is an easy option.

Fail 4 Maths a transferable skill

If knowing maths teaches you to think clearly, how come the world has been plunged into a financial crisis by people who are good at maths but couldn’t see the problems they were causing. The answer to this problem was identified by Thorndike over a century ago. ‘Transfer’, the degree to which learning transfers to actual performance in the real world is still a largely misunderstood or ignored issue in education. Learning is largely (not always) a means to an end, namely the application of that knowledge or skills, yet few educators know or care much about transfer. They assume it exists where it doesn’t (for example in maths and Latin) and make little effort to make sure it happens. Thorndike showed that transfer depends on the similarity of the situations or domains. This principle of ‘identical elements’ led him to recommend problem solving and practice in real-world contexts, so that the learning tasks and context matched the real world. Has this lesson been leant in the teaching of maths, or Latin? No.

FAIL 5 Calculators calculate

Almost everyone has a calculator in their pocket, as it’s a native app on almost every mobile phone and computer. Yet we insist on teaching people how to ‘calculate’ as opposed to useful, functional numeracy. Experts, like Wolfram and others, have pointed to the crude culture of ‘calculation’ in school maths, at the expense of real, functional and conceptual maths. Richard Norris has shown that maths in the workplace is intimately tied up with computers, spreadsheets and others forms of software. Yet maths and ICT are treated as two separate subjects. Isolating ‘maths’ in this way presents it as a purely abstract and often irrelevant subject.

FAIL 6 Miscalculation on teachers

Statistically, your child was, is or will be, almost certainly taught by someone whose knowledge of maths is rather poor. We know, with mathematical certainty, that primary school teachers have poor maths skills. The recommendation of the recent Government report into Maths teaching is a minimum B pass in GCSE before you’re allowed to teach the subject. This sounds like a bad joke until you realise that our children are being taught by largely primary school teachers with an absurdly low competence in maths. It claims that, “Almost all of those on primary PGCE courses gave up studying mathematics at age 16. So, by the time they taught their first classes, they had not studied mathematics to any meaningful level for at least six years.” Only about 2% of primary school teachers have a degree in science or any STEM subject. Another shocker is the fact that in secondary schools, “24% of all children in secondary schools are not taught by specialist mathematics teachers”. Read that again. Most maths is not taught by maths teachers or even by teachers with a solid grasp of the subject.

FAIL 7 PISA ‘standards’

The PISA results show plummeting performance in maths by our young people. The Chinese have screamed to the top. We’ll be an economy the equivalent of Bangladesh in a few years if we don’t get our maths scores up.  This is all baloney. A more detailed analysis of why PISA is wrong.

This is a common mathematical problem among politicians, employers, even so called experts in education. Our performance has remained stable. There is no ‘drop’ in standards. If you construct a league table, you can, mathematically, rise and fall in that table while remaining the same in terms of competence. That’s the problem with league tables – they create the illusion of winners and losers.

Gove is an English graduate with scant knowledge of maths and science. I know because I challenged him on a shared platform at the Tory Party Conference in Blackpool when he claimed that all schoolchildren should know that the orbit of an electron relies on the same force as the orbit of the planets around the sun! There were guffaws from the audience, so I suggested he needed a new example as the forces at work here couldn’t be more different (true story). He went apeshit but he was still hopelessly wrong. His EBacc has all the hallmarks of a PISA-led curriculum, far too academic, and exclusive. His greatest crime is to have moved the goalposts after goals have been scored. If you change the goalposts so dramatically and quickly, you simply condemn 85% of students as failures (only 15% currently meet the Ebacc standard). What’s worse, Gove is applying the measure retrospectively. This is like moving the goalposts at the end of the game and disallowing goals scored. It’s madness. Do the maths. You can have schools with high achievement in Maths and English plummet down the new league tables from near the top to near the bottom, as they haven’t focused on humanities or languages. One weird consequence is that a student who does Latin and Ancient History will be judged above those who do Business Studies, Engineering, psychology, a third science and lots of other subjects. It’s worse than bad, it’s perverse.

Conclusion

We don’t actually live in a more mathematical world. We live in a world where most maths is done by calculators, computers and machines, or a relatively small number of experts. The vast majority of us need little actual maths, other than ‘functional maths’. To funnel all young people into a path that demands a mostly irrelevant, maths curriculum is to turn them off school and learning. This obsession with maths may, mathematically, be the very things that lowers our general educational attainment.

Calculators: Education stuck in pre-calculator age

Archaeological evidence for an abacus goes back to 5th century BC Greece, however, there is indirect evidence of their use in Mesopotamia, Egypt and Persia. It is still widely used in Asia. The humble electronic calculator was the first computer to impact teaching and learning. It quickly replaced mechanical slide rules and mechanical calculators in the 1970s. Calculators now include scientific, algebraic, trigonometric  and graphing functions.

Education is still stuck in pre-calculator age

Everyone’s miserable about maths: employers, politicians, teachers and especially learners, many who fail and hate the subject with a passion. Indeed, governments have become obsessed with the subject, largely on the hysteria surrounding the PISA rankings.

One issue that is receiving intense attention is ‘calculation’, which is kicking up a storm in maths education. The ubiquity of calculators has led some to question the way we teach maths in schools. They claim that the world has changed from analogue to digital and the teaching of maths needs to respond accordingly.

Some argue that calculators have led to a reduction in numeracy and maths skills. They recommend not using calculators in schools until a certain level of competence in mental arithmetic is reached. Others argue that the traditional focus on ‘calculation’ needs to be replaced by a more sophisticated curriculum of solving problems using maths. Why teach long division, when you are unlikely to ever use it in real life? Calculators can also be used to do the necessary calculation spadework on algebra, trigonometry and graphics.

Maths need exaggerated

Some, like Roger Schank, believe that the need to learn maths is grossly exaggerated as only a tiny proportion of adults will use the maths that is taught, beyond basic arithmetic. His point is that most of what is taught, especially algebra, is of no real practical use and does not help people to think logically. He often asks highly educated audiences to tell him the quadratic formula – few ever answer. Sure, some will need maths in their later career, so says Roger, let them learn it later. Roger has traced this obsession with maths back to early 19^th century curriculum choices and claims that this is a historical problem, fuelled by the fact that maths is easy to test, especially ‘calculation’

Too much calculation

Conrad Wolfram decries the focus on ‘calculation’ in school maths. We spend most of our time teaching calculations by hand, which any calculator and computer can do better than any human. Practical, mental arithmetic is fine, but what are these numeracy basics? Automation pushes the user towards using the tools in more sophisticated ways. Maths is not calculation and over the last thirty years calculation has been automated by calculators. Education is still stuck in a pre-calculator age.

Far better to understand what you’re trying to achieve. He recommends that programming is a better way to do maths. It makes maths more practical and academic at the same time. He goes further and argues that the obsession with calculation in maths kills off the initiative, intuition and perseverance that maths needs. In other words we’re turned off maths by maths. Students learn to look for and apply formula, which they then proceed to calculate. Text books are full of primitive, dry, exercises that seem like chores. Many now argue that real life problems should stimulate mathematical enquiry through the use of more word based problems.

Calculators and computers

A calculator is pretty standard as a native application on PCs, Macs and mobile devices. Tills automatically calculate the correct change for customers. Calculators are therefore embedded in newer forms of technology making them more readily available. This is one potential use of mobile devices in schools that teachers should consider.

Conclusion

Maths is forced, by law, upon people who see it as lacking relevance and don’t want to learn it, taught by people who, because they’re good at maths, often don’t know how to teach it. Yet the curriculum is aimed, largely at those very few who will use high-level maths professionally.

Why are the Maths Zealots in our schools?

There’s a movement stalking our land and many other lands – the Maths Zealots. Driven by PISA envy they are zealots, sure that the solution to all our economic and social woes is knowing the quadratic formula. Before I start, don’t shoot the messenger. I like maths, I’ve taught maths, I helped commission major maths teaching projects. I just don’t like silver-bullet movements that take one subject and treat it as if it’s the pinnacle of educational achievement, when it’s clearly not. Let’s be honest, the last person you’d take most problems in life to, even practical ones, is a maths teacher or prof.

Latin - educational fossil

This is not new. Latin was forced down the throats of generations of kids for no better reason than laziness and snobbery. No it’s not a sensible way to help you learn other languages. No it doesn’t give you great insights into English. It’s a long-dead fossil that bores most with mostly fictional educational benefits.

Coding the new Latin

I have no doubt that coding should be taught in schools but compulsory coding for all is crazy. I’ve coded, spent a lifetime working with ‘coders’ and most of them will say that it’s not a subject for all. I don’t care how many ‘hackathons’ you’ve organised, coding is a specialist skill, readily purchasable from smart people in cheaper lands. For every coder there’s project managers, sales people, marketing people, finance people, graphic artists, video production….. funny how you never hear of the ‘hour of sales’, even though it’s the skill that’s most often absent in start-ups.

STEM - all too easy acronym

Some acronyms create more problems than they’re worth and this is one of them. I’m OK with promoting science but the ‘T’ is only there so that SEM is easier to say and remember. Is it “Technology or ‘Technical’? Doesn’t matter – what is the ‘T’ bit? Engineering? Isn’t that technology? Not sure. But what has happened – the design bit of ‘E” for engineering has been ripped out of the school curriculum by Arts graduates like Gove. Then let’s just lob maths onto the end. let's add some more letters like 'A' for the arts, "D' for drama and design, 'H' for history, 'G' for Geography, 'L' for languages, 'B' for Business, "P" for psychology, 'M' for Music and so on. Then again it wouldn't fit the great desire for all too easy to remember acronyms. STEAMBLGPHD...?

Numeracy

To be fair this is a more appropriate approach to maths but as is so often the case, rather than argue the case for a functional maths qualification, the quangos and charities turn this into a maths crusade. People are so focused on getting their MBEs and CBEs that they forget that this is primarily a political battle, where politicians and civil servants have to be challenged.

Maths, maths and more maths

More maths is likely to cause more problems. Why?  Remember what Henry Ford said, “If I had asked people what they wanted, they would have said faster horses.” Hammering home more calculation maths is a case of trying to shut the proverbial stable door one the horse has bolted. We desperately need a functional maths qualification, based largely on number theory but not just on calculations. Computational devices, we all have one, it’s called a mobile, do the job better. If we are to teach maths it needs to be more focussed on the practical and mathematical thinking. The GCSE maths standard is crap. It even has maths mistakes in the specification. Algebra, Trigonometry ad most of Geometry is neither necessary nor desirable for most learners. The vast majority of people in the workplace and in life do NOT need even GCSE maths. We have calculators, some basic number theory and functional maths will suffice. Even worse is the idea that you need GCSE maths in every apprenticeship and vocational qualification. This is patently false and is likely to result in massive and unecessary failure rates.

Conclusion

The myopia produced by single subject advocates so often descends into ugly zealotry. I have only defriended two people on Facebook, ever, and one was a persistent grammar and spelling nut who confused typos with stupidity. She never said anything remotely interesting on any subject other than police posts for spelling and grammar errors, even then she was often wrong, confusing Latin rules with English. I see the same behaviour in the single-subject nuts, who want to funnel us all into a small set of subjects. Rather than look at the breadth of subjects and open up young minds to many possibilities, they want to close these young minds down into human calculators. The common denominator here (to get all mathsy for a moment) is their desire to get the rest of the world to think like them. What THEY learnt, everyone should learn. I’ve met enough of them to know how catastrophic this would be.

PS

To show that I’m not against maths and computer science per se, check out Citizens maths and OCRs Computer Science MOOC. Two projects I'm proud to have helped get started.

AI maths app that students love and teachers hate

We’ve all been stuck on a maths problem. Look up a textbook – hardly ever helps, as the worked examples are rarely close to what you need and explanations clumsy and generic. What you really need in help on THAT specific problem. This is personalised learning and an app called Photomath does it elegantly using AI. Simply point your mobile camera at the problem. You don’t even have to click. It simply scans and comes up with the answer and a breakdown of the steps you need to take to get to the answer. It can’t do everything, such as word problems, but it’s OK for school-level maths.

Getting there

The app is quite simple at the moment and only solves basic maths problems. It has been criticised for being basic but it’s at this level that the vast majority of learners fail. But it’s getting there and I don't want to get hung up on whether Photomaths is as good as it says it is. or better than other maths apps. For me, it's a great start and a hint of great things to come. In fact Wolfram Alpha is a lot more sophisticated. But it is the convenience of the mobile camera functionality that makes it special.

The problem that is maths

Maths is a subject that is full of small pitfalls for learners, many which switch off learners, inducing a mindset of ‘I’m not good at maths’. In my experience, this can be overcome by good teaching/tutoring and detailed, deliberate feedback, something that is difficult in a class of 30 plus students. This subject, above all others, needs detailed feedback, as little things lead to catastrophic failure. This approach, therefore, where the detail of a maths problem is unpacked, is exactly what maths teaching needs. It is a glimpse of a future, where performance support, or teacher-like help, is available on mobile devices. AI will do what good teachers do, walk you through specific problems, until you can do it for yourself.

Students love it, teachers hate it

Predictably, students love this app, while teachers hate it. This is a predictable phenomenon and neither side is to blame. It happened with Google, Wikipedia, MOOCs,…..  and it’s the same argument we heard when calculators were invented. The teachers’ point is that kids use it to cheat on homework. That depends on whether you see viewing the right answer and steps in solving an equation as cheating. In my opinion, it simply exposes bad homework. Simply setting a series of dry problems, without adequate support, is exactly what makes people hate maths, as help is so hard so find when you’re sitting there, on your own, struggling to solve problems. Setting problems is fine for those who are confident and competent, it often disheartens those who are not.

Sure the app will give you the answer but it also gives you a breakdown of the steps. That’s exactly where the real leaning takes place. What we needs is a rethink about what learning and practice means to the learner (and homework) in maths. The app is simple but we now see technology that is, in effect, doing what a good teacher does – illustrating, step-by-step, how to solve maths problems.
Homework
Homework causes no end of angst for teachers, parents and students. Some teachers, based on cherry-picked evidence or hearsay, don't provide any homework at all. Many set banal and ill-designed tasks that become no more than a chore to be endured by the student. I personally think the work 'homework' is odd. Why use the language of the workplace 'work' to describe autonomous learning? In any case, we must move beyond the 'design a poster'  and get the right answer tests, to encoring autonomy in the learner. This means providing tasks where adequate support is available to help the learner understand the process or task at hand.

AI in learning

AI is entering the learning arena at five different taxonomic levels; tech, assistive, analytic, hybrid and automatic. This is a glimpse of what the future will bring, as intelligent AI-driven software delivers, initially assistance to students, then teacher-level functionality and eventually the equivalent of the autonomous, self-driving car. It's early days but I've been involved in projects that are seeing dramatic improvements in attainment, dropout and motivation using AI technology in learning.

WildFire

I’ve been using AI in a tool called WildFire that uses semantic AI to create online learning content from ANY document, PowerPoint or video. No lead time, sophisticated active learning and a massive reduction in cost. We’re starting to see a new generation of tools that use smart AI techniques to deliver personalised learning. AI is fast becoming the most important development in the advancement of teaching we’ve seen to date.

Vorderman on maths – reactionary TV presenter, no maths degree, debt & property scammer advises us on maths!

We’ve had an endless stream of ‘I’m a celebrity, let me fix your schools’ types this year; Jamie Oliver, Toby Young, Joanna Lumley, and now, god help us, Carol Vorderman. (Interesting to note that this Conservative supporter wouldn't be allowed to teach maths, as Gove doesn't want teachers with anything less than a 2.2 - she has a third.)

Vorderman – a few unsavoury facts

Just a few words about Vorderman: a) She doesn’t have a maths degree, she has a third class degree in Engineering, b) She acted as a spokesperson for the rogue debt consolidation company First Plus, forced to cut the contract after criticism from the debt charity Credit Action c) She fronted a property company that collapsed leaving many with unfinished properties abroad which they had paid for, d) Sacked from Channel 4 after being seen as a money-grabbing lightweight on £1 million a year, e) After a disastrous appearance on question time, where she spouted extreme right-wing views, Dimbleby said in the Times, “It lasted an hour, this programme...it felt like more to me.” f) she also has a long history of being partisan on educational politics and attacking the Labour Party.

So let’s imagine the following conversation at Tory Party headquarters, who commissioned the report when they were in opposition; “Suggestions to sort out maths in schools? How about Carol Vorderman? Does he have a maths degree? Well no, and we’ll have to hide that fact that she’s encouraged dodgy debt management, fronted a failed property scam and spouts reactionary nonsense whenever possible. But, she does have one redeeming feature. What’s that? She’s ‘rear of the year’. Call her.”

To be fair, apart from Carol, the team is academically sound, and has made some interesting observations and recommendations.

Curriculum

They conclude, that the maths curriculum is a catastrophic, irrelevant mess, geared towards higher advanced maths at the expense of functional maths. I couldn’t agree more. Teaching 14 year olds how to use the quadratic formula and surds is just plain stupid. Roger Schank often asks his academic audiences whether any of them can remember the quadratic formula, and he rarely, if ever, gets a correct answer. Why worry then that, “Only 15% of students take mathematics, in some form, beyond GCSE” as the current GCSE is hopelessly geared towards high-level, irrelevant, abstract maths. I think 15% is reasonable, if not a little high. And if “Nearly half of all students ‘fail’ GCSE Mathematics, why worry, as it’s a flawed, overly-academic and partly irrelevant qualification.

The GCSE curriculum is loaded with esoteric algebra, trigonometry, geometry and number theory that 99% of learners will never, ever use in their entire working lives. Note that this is at the expense of functional maths in two senses, 1) it squeezes practical maths out of the curriculum, 2) it is a massive demotivator, reinforcing the idea among millions of children that ‘they can’t do maths’.

The suggestion that we have a mainstream Maths GCSE that focuses on functional numeracy is therefore wise. This is what I had at school in Scotland many moons ago. I did an O-level in Arithmetic (practical) and another in Maths (theoretical). Makes sense, although I’d reframe Arithmetic as ‘Practical Maths’. Employers aren’t complaining that people don’t have ‘maths’ skills, they’re complaining because they don’t have basic ‘functional numeracy’.

Teaching

At one end of the spectrum the team are spot on – primary school teaching. The teaching of maths at this level is woeful; mostly because the vast majority of teachers have very low numeracy skills, and partly because of poor teaching methods. In the same way that whole word teaching had a catastrophic impact on literacy; ill-informed, half-baked, non-integrated and inconsistent approaches to numeracy teaching have also been catastrophic. There is the recommendation that the teaching be rooted in the real world, through practical tasks – something that’s been recommended for decades but been studiously ignored in schools.

Almost all primary teachers stopped maths at 16

The recommendation of a minimum B pass in GCSE in maths before you’re allowed to teach the subject sounds like a bad joke until you realise that our children are being taught by largely innumerate primary school teachers. It claims that, “Almost all of those on primary PGCE courses gave up studying mathematics at age 16. So, by the time they taught their first classes, they had not studied mathematics to any meaningful level for at least six years.” Only about 2% of primary school teachers have a degree in science or any STEM subject.

Most maths not taught by maths teachers!

Another shocker is the fact that in secondary schools, “24% of all children in secondary schools are not taught by specialist mathematics teachers”. Read that again. Most maths is not taught by maths teachers! However, the team have fallen into the trap of seeing the solution to bad schooling as yet more schooling. Forcing young people to study maths until they are 18 is just plain lunacy. If you haven’t got basic, functional numeracy into your head after 11 consecutive years of maths, another two years isn’t going to matter and the idea of ‘maths citizenship’ is just weird.

Conclusion

The report points out 1) the people teaching maths are by and large amateurs, 2) the curriculum is too esoteric, 3) we need two separate maths qualifications. I agree with all of these findings but we’re chasing moonbeams here. First, the educational establishment is so wedded to dated PGCE recruitment and curriculum practices that it is almost impossible to reform without radical restructuring. You have to get teacher training out of the Universities where it reinforces the old academic model and change the methods of recruitment. Secondly, you have to break the back of the gold standard, A-level mindset, where University entrance is the primary goal of all schooling and everything else is classed as failure. It ain’t going to happen.

Download full report here.

7 reasons why kids right to have gone apeshit over this GCSE maths question

A maths question posed to 16 year olds in England caused a Twitterstorm, as teenage wrath came down upon the examiners for being deliberately obscure, oblique and irrelevant. A petition has reached 15,000 signatures (impressive number), asking for the grade boundaries to be changed. Something freaked these students and it's worth reflecting on what could have gone wrong. After all, in previous exams, impossible to answer questions have been set.

The maths Taliban would love every child to be able, like them, to solve this conundrum and their reaction has been to blame the students and claim that here's nothing wrong here. But i think there is fire behind this smokescreen - it is a poor test item. It is also a touchstone for educational debate.

1. Bait and switch

Although it sets you up with a probability question, they bait you with probability, then switch to algebra and throw in a quadratic. To understand why this is a problem you have to understand that the maths GCSE curriculum is taught in silos – number, algebra, ratios, geometry, probability and statistics. Maths is not (or rarely) taught in terms of the broader mathematical thinking skills required to answer this type of question. Kids are taught to identify the ‘type’ of question in relation to the curriculum categories they’ve been taught and apply the rules they’ve been taught in that one domain. I understand that this is not right but it's the reality. That, in iteself, I suspect, explains the uproar.

2. Final probabilty is given to problem

Rather puzzlingly, they use the all too predictably infantile, Beano-like, how many sweets are in the bag (really - sweets?) story. This is a bit pathetic and an artefact that seems to only exist in the minds of exam setters. They are then confused by being given a known probability (1/3). This sets up dissonance in the mind of the learner. They think - hold on – I though we were being asked to find out how many sweets are in the bag (solve a probability problem) now I’m being given a final probability – that doesn’t make sense. By giving the final probability in the question, you confuse the student. They’re asked to fill out the middle of a sandwich, which is rarely real, and not what they’ve been doing for the last four years. In fact it’s rare in mathematics. I suspect that a lot of students would have been panicked by this on first reading.

3. Trap

As an assessment item it is poor, as it lays a trap for students. The appearance of a quadratic equation in the question n² – n – 90 = 0 suggests that this needs to be solved - that's almost what asked for in the final statement. Most of the young people sitting this exam would have been drilled in solving equations like this. Again, I'm not saying this is right, but it's how it's done. This was part b) of the question but many would have been thrown trying to answer part a) and not got that far.

4. Show what?

Show what? Another problem lies with the word ‘show’. What does that mean in maths? Draw a diagram? Solve the equation? Come to a conclusion? Provide a proof? It’s a vague term that should have been made much clearer. I know this is common in GCSE papers but more exact instructions would help. For a good discussion of the 'show' issue in maths see this TES piece.
5. Position
The question was not at at the end of the paper but earlier than expected for such a difficult test item (19/25). Students are taught that the questions ramp up in difficulty, with questions like these right at the end. By appearing relatively early in the paper it would have stumped many and lost them time and confidence for the completion of the last six questions. This, for me, is the one reason the grade boundaries should be reconsidered.

6. Irrelevant

I’d set a high probability on the fact that the majority of politicians, supporting the maths obsession, and people working in the Department of Education, wouldn’t be able to and don’t have to be able to complete this problem. I’d say with huge certainty that the vast majority of adults, functioning well in their lives and jobs, would not be able and don’t have to be able to complete the problem. The vast majority of adults will never have to do algebra anywhere near this level of sophistication in their lives. I’d go even further and say with a very high probability that no human being has ever thought of this particular exercise as a problem, never mind used it in any practical sense. Let’s suppose we put relevance issues to one side, it is still a poorly designed test item.

7. Non-functional

Even those who do go on to higher education do not need to be able to solve this problem. In fact, the question may have skewed the Edexel exam away from the other boards Maths exams, unfairly hitting those who just happened to find that their school went with this exam board. This is a serious weakness in the English education system. We have been driven towards abandoning vocational skills, which largely need ‘functional’ maths, not the obscure algebra embedded in the GCSE curriculum. We are now locking students into a ‘resit until you pass’ model that does nothing more than make them feel like failures and turns them off progression in their education. The sooner we have a functional (not foundation) maths qualification, alongside the GCSE the better.

Maths in schools

The issue should be made clearer when the stats emerge for student performance on this question and the paper as a whole. I’ve taught maths like this to teenagers who don’t see the point of it all, and they have a point. First, as I have argued, this question raises the question of the quality of our exam boards and examiners. We have had impossible to answer questions and although this is just a badly constructed test item, quality control is clearly an issue. More importantly it is a symptom of something far worse – our modern obsession with maths in schools.
Quality

We really do have to ask whether we need multiple examination bodies in England. It increases the probability of mistakes being made and make no mistake, they are being made. This is not a one-off incident. AQA this year posed a chemistry question, then gave the full answer just a few questions later. We have had OCR ask and impossible to ask Maths question in an AS paper. The question as printed asked candidates to verify the shortest route, for two given conditions, giving values of 32.4 + 2x km and 34.2 + x km. These values should have been 34.3 + 2x km and 36.1 + x km respectively. The error was not to have included twice the journey between A and B (0.9 km) and the journey between F and G (1.0 km) in the values given. More errors were found found in A-level physics and a GCSE Latin paper from the OCR exam board and an AQA maths GCSE foundation paper. AQA admitted they had another rogue question in this year's A-level paper. Ofqual have proven to be toothless, all too ready to defend the hapless examiners and examination boards, when students were clearly affected by these errors. The quality control is clearly insufficient, realt tests by real students is nor rigorous enough and poof-checking lax.

Solution

1. Express two probabilities:

Hannah takes the first sweet, so there is a 6/n chance it will be orange.

(6 oranges and n sweets)

Hannah takes a second sweet, so there is a 5/(n-1) chance it will be orange.

As there are only 5 orange sweets left out of a total of n - 1 sweets.

2. Calculate two probabilities as one:

The chance of getting two orange sweets in a row is the first probability multiplied by the second.

Which is 6/n x 5/n–1

3. Express as a single equation:

We’re told that the chance of Hannah getting two orange sweets is 1/3.

So: 6/n x 5/n–1 = 1/3

4. Rearrange equation:

 (6x5)/n(n-1) = 30/(n2 – n) = 1/3

Or 90/(n2 – n) = 1

So (n2 – n) = 90

Hence: n2 – n – 90 = 0