A-Level Maths guide

A guide for independent learners on how to study A-Level Math, based on everything I learnt along the way

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Introduction

I unexpectedly received an A* at A-level Maths with Cambridge International, after taking 4 exams across two winters. I didn't pay for any tutoring, study A-Level maths at school, retake any official exams, pay for re-marks, etc. My main method was simply to study every single day, working my way thoroughly through official textbooks, and, closer to the exam, complete an abundance of past papers (all of them). For comparison, whilst a teenager, I scored C, E, F after losing interest and motivation.

This webpage is about just providing everything (something akin to metadata) that I learnt about studying and A-Levels along the way; it won't cover any specific math topics, only the things you should know before and whilst approaching them.

Can you, personally, study A-Level Maths independently?

I received an A grade at Maths GCSE and was top of the class as a child in infant school, alongside a British-born Indian friend. My GCSE teacher did actually encourage me to take A-Level maths but I naively chose otherwise. After leaving school, I enjoyed delving into applied kinds of maths independently, such as sudoku, and programming (programmed a couple sudoku solvers, even).

I don't say the above to boast, but I am trying outline the fact that it probably takes some natural propensity and also interest, if you are to succeed. If you have naturally engaged in any kind of heavily numerical or logic-based activities, be it puzzles or other hobbies, and at any point excelled in maths whilst at school, it seems your potential is good. If you have not, you would be very much in unchartered territory.

In the words of 'David Game', headmaster of a college he founded, now in central London: 'really the attitude of the student is the most important thing'.

Something else to note about maths is it is fortunately less memorisation-heavy than other 'english comprehension' subjects that emphasise collecting knowledge, such as History, or even Computer Science.

Exam boards & availability

Exam boards vary in a much bigger way than I initially realised, and it is important to choose carefully.

The exams of the 3 major exam boards (AQA, Edexcel, OCR) which are regulated by a government organisation, Ofqual, have, since 2017, undergone an overhaul of their syllabuses and are now only avilable as 3 written papers, with different question styles compared to the past. As a result, there are very few past papers of the new style available, outside of textbook unofficial versions. This makes revision more difficult, and less targeted.

Unfortunately, there is now also only one period of time you are allowed to take exams from these 3 exam boards - summer (complications caused by pandemics were never considered) - compounding the lack of past papers, and so you may need to plan around this, and find somewhere with a tolerable room temperature. If you opt against having tutoring, exam fees are generally the biggest expense.

Then there is Cambridge International, an exam board which are a part of Cambridge University, which are able to do things quite differently, more desirably, as a result of being independent, by mainly focusing on the international market.

People who do not attend a school and register for exams independently are colloquially known as 'private candidates'. There are colleges (you sit alongside teenagers) and exam centres (for-profit businesses) who admit private candidates in their exam rooms on exam dates, in exchange for a fee. You may need to do alot of research and make alot of calls, to find where you can take which exams, for how much, and when. The exam board websites tend to have features which show colleges accepting private candidates, but this is to be treated as only a starting point, as they are often very out of date. You will also have to take responsibility for yourself regarding exam board deadlines, entry codes, and communicating with exam officers (not too much if you opt for a college, as you won't be their primary customer, but not too little either). All exam boards publicly release dates about registration deadlines online. With CAIE you should receive your 'statement of entry' (a document containing your details and registered exams) before the 'final registration deadline' (after which a registration may be denied by CAIE). Other exam boards may differ.

All exam boards post the details of their syllabuses on their respective websites, which often include the formulae that will be provided during the exam, which you should definitely have a copy of when studying, so you know what not to memorise. Note that different exam boards may provide very different formulae, some more generous than others on certain topics.

Edexcel

This is a run by the for-profit, FTSE100 company 'Pearson', with offices in central London, and has the largest market share for A-Level Maths exams. As a result there are usually many places for private candidates to take an Edexcel exam. Pirate copies of the digital versions of their textbooks (and Edexcel-focused textbooks from other providers) are freely available online, whilst second-hand physical copies are sold cheap after exam-season. Official full solutions are freely available online for every question in the official Pearson textbooks. There is a giant Casio logo in the initial pages, and re-marks cannot be requested per paper, only for the full qualification (potential unexpected fees, perhaps due to loss of profits from AS Levels).

There are 2 pure math exams (66% of total grade) and 1 applied math exam (statistics and mechanics). The total fee, in London, is approximately £200 or upwards.

OCR

This is the UK oriented exam board which is run by Cambridge Assessment, a not-for-profit organisation which is a part of the Cambridge University. The textbooks are more expensive, but more luxurious in typography and design, with good accuracy of answers. Full solutions to answers are available with digital editions. Exam availability is significantly lower than Edexcel and AQA, and, as a result of supply-and-demand, OCR is a more expensive option, both exams and books.

Similarly to AQA and opposite to Edexcel, there is 1 pure math exam and 2 applied math exams (statistics and mechanics), each worth a third of total grade. Presumably one book per exam, more desirably. Fee, in London, is approximately £400.

AQA

This is more a not-for-profit exam board, that I investigated last, when I got to Further Maths. The availability in terms of exam registration is generally equal to Edexcel; frequently both are offered to private candidates.

The AQA Maths and Further Maths courses have the following upsides: same quality publishers and authors as OCR, only 3 exam dates (Further Maths with other boards requires 4 exam dates, which can cause clashes, if you are doing other A-Levels). For example, the textbooks from Hodders and Cambridge University Press are AQA-oriented rehashes of their OCR equivalents, and so this potentially means the content of OCR and AQA are more closely related than to Edexcel.

Downside of AQA Further Maths I found are that, if you choose the Discrete Further Math module, there is no help on the formulae sheets, nor with Edexcel. In contrast, OCR does provide short guides on implementing the different major algorithms, saving memorisation. Furthering, some questions in the exam (e.g. drawing transformations of ellipses) are distinctly biased towards graphing calculator users.

There are 3 exams, of which 2 are applied math (statistics and mechanics, separate), each worth a third of total grade. Fee, in London, is equivalent to Edexcel.

Cambridge International - CAIE

This is the other exam board run by Cambridge Assessment. Their main focus is on the international market, and so they are not subject to the whims of the regulator Ofqual. However, as a result of being run by Cambridge Assessment, universities treat them equally. At an open day, headmaster David Game proclaimed that Cambridge International IGCSE exams are harder, but allegedly Edexcel GCSEs have become harder since 2017, according to the Edexcel syllabus.

A major advantage of CAIE is that there are two exam seasons per year - the first in summer, the second late autumn. This, combined with the fact you can split half the exams at different times (1 pure and 1 applied in summer), makes it easier to begin your studies anytime, and also have some affirmation as to how you are doing, at the half-way point. Doing your A-Level in two separate halves, across two seasons, also creates the unique mentality of 'having something to lose', which can be an added motivation. It also reduces memorisation, by staggering your workload.

Further major advantages of CAIE are that, as they are not regulated by Ofqual, their exams have not been forced to change so suddenly since 2017, and so the past papers are still relevant to the current exams. Plus, because they provide exams to different parts of the world (with different questions), there is an enormous abundance of past papers to practice on.

The quality of tutorials in the CAIE official textbooks is usually good, however, a disadvantage of them is the questions/answers are not of the same quality as OCR or Edexcel - answers are wrong or proofs impossible in significantly higher frequencies, and it takes a certain amount of time, self-confidence, and perseverence to get to the bottom of these errors, as full solutions are not always published. Perhaps this is a consequence of catering to a less profitable international market. On the other hand, multiple publishers (commissioned?), such as Hodders, Nelson, Collins, create their own series of CAIE textbooks, which are endorsed by CAIE, and I have found Hodders' series to be particularly good. L. Bostock's Mechanics 1, published by Nelson, was also a great supplementation for me when starting out.

Note however that CAIE do iterate smaller changes to their syllabuses over the years, and so the older textbooks are replaced, with content changed in some ways (e.g. for 9709 Maths: no more vector planes, no more Mechanics 2, but still a noticeable quantity of errors in answers). The latest CAIE textbooks have much less written english (Douglas Quadling often went on whimsical, interesting tangents), potentially to suit an international audience.

There are 2 pure math exams (60% of total grade) and 2 applied math exams (a choice of Mechanics 1 and Statistics 1, or Statistics 1 and Statistics 2). The fee, in London, is approximately £300 for 2 exams (half A-Level), or £400 for 4 exams.

Subjective exam board detour

I started studying April 2019, with Edexcel's 'Pure Math 1' textbook, which was recommended to me by aforementioned GCSE maths teacher, and I found it a good book to start with due to the full solutions, as I had forgotten a large portion of GCSE maths. However, as the soonest exam available was then CAIE, after completing Edexcel Pure 1 back-to-back I then moved onto the CAIE Mechanics 1 textbook by Douglas Quadling, which felt significantly harder (note: the Douglas Quadling series is somewhat out of date since the beginning of 2020). After finishing that back-to-back, and with autumn exam season approaching, I then purchased Quadling's CAIE Pure Math 1, and was shocked to find the content very different to Edexcel's Pure 1 (I later discovered it is the orders of the topics across each of the exam board's series of books that differs). I was then studying extremely hard, and without the desirable level of depth, to make up the differences, learning many, many new concepts (reverse chain rule, radians, arithmetic & geometric sequences, 3D vectors, reversing functions) not long before the exam, and was not able to complete an optimum quantity of exam papers as a result. However, this experience did teach me a big lesson: I could be studying much more intensively than I previously was. and so after exams, I did...

I booked a CAIE AS - Mechanics and Pure Math 1, worth 50% of a full A-Level. Ironically, there was only one mechanics question I floundered on, in the actual exams. I was pleasantly surprised to be awarded an A (approximately 85%, if I remember correctly) for the CAIE AS, and as a result then stuck with CAIE, doing only CAIE textbooks until the next exam season.

I registered for Edexcel exams alongside the second half of the CAIE A-Level exams, as I felt I needed a backup, as a private candidate. I was expecting to take CAIE, then a few weeks later, Edexcel. but they were all cancelled due to the Covid-19 pandemic. I did fit in approximately a month of CAIE Math revision in the summer.

I was progressing with the CAIE Further Maths textbooks by the time the autumn exam season came around, when Edexcel exams were suddenly announced to be available for October, just a little over a month beforehand. I put in only 2 weeks of Edexcel-specific revision. There were numerous gaps in my paper as a result (things I hadn't looked at since 2019, or Edexcel-specific bits). But despite expecting an A or B from Edexcel, I was pleasantly surprised to receive a mark of 199/300. 214/300 is an A*, 164/300 is an A, so you could call it an A+, if you split the boundary into three.

I was originally planning to only do CAIE 9231 'Further Maths' AS, a separate subject to CAIE 9707 standard Maths, in autumn, but ended up taking it alongside the second half of my CAIE standard Maths exams. This fit relatively nicely, however, as I was able to do both Statistics 1 and Further Statistics, at the same time. Before 2020, CAIE A-Level Futher Maths papers had to all be taken at once, which I could not have been prepared for at that point. I scheduled my past papers carefully, and completed approximately all the CAIE past papers that have ever been released (approximately 50 per paper for 9231 Further Maths, and 75 per paper for standard 9707 Maths, but no 9709 Statistics 2 papers). This, after completing all the official textbook questions in the latest Pure Math 2, Statistics 1, Statistics 2 textbooks, plus the Further Statistics and Further Pure Math 1 parts of the official Further Maths textbook (honeycomb cover).

Overall I felt the Edexcel exams to be disorganised - despite there being 4 textbooks, 2 being pure math, 2 being applied math, and 3 exams total, the content of *both* the pure math textbooks is now being wildly jumbled across the 2 pure math exams. This is in sharp contrast to CAIE, and potentially also AQA, OCR, where one textbook correlates to one exam. I felt that this was a foolish decision by Edexcel, a crude way to make something harder just by making it as unpredictable as possible. For example, if we iterated this idea out, it is akin to walking into an exam and not knowing whether it will be an English or a Maths exam. Preparation is forced to become more untargeted. Furthering, a few too many questions were essentially a regurgitation of proofs. Two proofs by contradiction (across both papers, naturally) - the most Alice-in-Wonderland proof you will find in the latest syllabus - and one proof of the sum of an arithmetic sequence. Meanwhile, graphing calculators are allowed, which have more than enough storage for these proofs... However, I do admire the proof from first principles, which has cropped up in a papers since 2017, but which unfortunately did not for me. I was surprised to even find an arithmetic series question in the Statistics paper. There were also disappointments such as marks awarded for memorising something about a data set I had never looked at, or presenting an answer in a fancy format ('set notation'). As a result, my secondary option for Further Maths became OCR (and later AQA), and I eventually sold my Edexcel Further Math textbooks (excluding 'Decision 1', which I completed) and purchased OCR instead.

CAIE results are released at a later date, and I was very shocked to have received 91%, an A* for standard maths, and 94% for my Further Maths AS.

I went on to complete all the questions in a variety of different textbooks for CAIE: Hodders F. Pure 2, Collins F. Mechanics, Quadling Mechanics 2, Nelson Mechanics 2, the entirely of the CAIE Further Maths official textbook. Plus OCR F. Pure 2 and F. Mechanics from Cambridge U. Press.

Due to an exam clash with some non-Maths exams, and Ofqual exams again being announced only two months before, I switched from OCR to AQA, and was pleasantly surprised to find the textbooks for AQA and OCR by Cambridge University Press were largely a re-hash - same authors, mostly same content - differing only on the 'past paper' questions they had included, and exam-board specific topics. I also did their Discrete Maths textbook, alongside some Hodders AQA Discrete math. However, I did trip up on a few parts of AQA-specific content in the exams, both the Pure and the Discrete. But my AQA Further Mechanics was pretty strong. Afterwards, I took the last two CAIE Further Maths exams and felt I only stumbled on one question, at the end, where I mis-interpreted a question and took a long route. As usual, I was awake at night reflecting on my answers.

Calculator

I changed calculator model once, during my studies, and I only wish I had changed earlier. Your calculator is extremely important for checking answers, particularly when learning independently, but equally in an exam setting. You need to thoroughly go through the manual, and work out exactly how different features can help you with different topics. For example, a numerical differentition should almost always be done with a calculator, to check, after integrating by hand, and compared to the output of the original function, with the same numerical input. Usually they match up to the majority of decimal places, and after a while you can instinctively tell when they do not match to an expected quantity of decimal places. I copied this idea from 'test-driven development', which is found in the world of programming. There are many other checks that you can do, including something similar to the above, for differential equations.

The large exception to the above is mechanics. Mechanics is heavily conceptual, with contexts that vary widely, and a heavy focus on the invisibility of forces. There are likely to be few calculator features to help, in this area.

One thing I have apprehensions about is not having changed calculators before undertaking Further Math, as the one I had become accustomed to could not handle complex numbers ideally: neither Euler format, nor indices larger than 3 or less than -1 (unless stacked with brackets). On the plus side, the simplicity of this calculator meant I only changed batteries twice (only once due to low battery warning) over the space of 2.5 years, and it never failed me in an exam, despite bringing a backup.

CAIE do not allow graphical calculators in exams, probably to even the playing field for international students in less materialistically abundant circumstances, but perhaps also because modern calculators are becoming packed with dubious features. As a result I did not use a graphical calculator whatsoever, But I did end up spending significant time, overall, drawing graphs using 'table' mode, whilst learning. It is unclear if this has been a disadvantage or advantage, overall.

Supplementary books

One of the most important books to supplement any exam board textbook is 'A-Level Math for Dummies' by 'Colin Beveride'. The author has worked as a maths tutor, and has thus made this a critical book for filling in all the gaps between GCSE and A-Level, as well as those caused by the official textbooks lacking in their explanations.

For applied math, things can become very conceptual, and less clearly numerical, such as forces, or permutations. I found it very helpful to read through textbooks by alternate providers catering to the same exam board, who approach the syllabus in different ways. Plus, they can provide additional practice on the newer syllabus material.

Occasionally I did look online for extra resources, but overall I never found anything that was as cleanly put together as textbooks. However, I did learn about resolving forces on an obscure website, though this may be better explained in the latest edition of the CAIE Mechanics 1 textbook.

Day-to-day-studying

You generally want to put in as many hours as possible, working thoroughly through the textbooks from back-to-back, if you want to maximise your grade. I would recommend a minimum of 3 or 4 hours per day, a minimum of 6 months before any exam, and I wouldn't guarantee an A on that. At least, these are the number I started out with, and I later found that this quantity of hours did not feel enough.

I'd recommend buying exercise books, rubbers, refillable pencil leads, all in bulk. You need a comfortable chair, a big enough table, and an environment free of distractions from electronic devices. You will need somewhere quiet, and a certain level of financial stability so that you can focus primarily on your objective. Prioritise sleep and rest.

Your main competition is teenagers, some who are privileged to receive instruction from people who know exactly what they need to learn (without anything superfluous) and who know where mistakes are commonly made, and other students who are not at all interested in actually studying! Perhaps, like me, you were once one of them (the latter). Your main advantage is time, as school students, whether over or underachievers, are extremely time-pressed, their lives flooded with distractions from their school, parents, friends, crushes, internet businesses, etc. The priorities of young people can be fascinatingly misaligned. I've seen young people applying lipstick before an exam, scrolling through social media feeds, forgetting calculators, oversleeping, showing up late, missing exams.

Note that, at some point, your circumstances will never be quite perfect, and you will have to find some ways to manage. For example, whilst practicing a math question on the bus on the way to my first A-Level math exam, I accidentally missed the stop (windows steamed up, hadn't done the journey much) and ended up 2 stops down a dual carriageway. I tried thumbing a ride back, but no one cared. I caught a bus back halfway but then there was a massive traffic jam in the direction I was now travelling. The driver said "something's happened". So I was running down the street along some houses, much longer than I had run for a long time... There was even an announcement before I missed the stop, that the only train I could have taken instead was undergoing an outage. Wisely I had left 2 hours before the exam began, and made it early. But if I had missed that exam, then missed the next summer's during Covid-19, maybe I'd have given up, without ever knowing my potential... Then there are noisy neighbours, family wanting favours, people with contagious airborne diseases (incl. before Covid-19!) etc. - find ways to engineer these problems away.

Self-belief, self-honestly, health

Without descending into some kind of unbearable self-help book, I must touch on the importance of self-belief. Doing your A-Levels totally alone is unconventional. If you share this decision with others, there are people who will initially be supportive, who later become skeptical, and others who are skeptical from the outset. There aren't many other types of people! There might be a correlation between a person's own skepticism and their level of education - avoid falling into this trap yourself. Keep an open mind, give meritocracy a try with everything you've got (with a good exam board), then draw conclusions.

At the same time, self-honesty is very important. Cover up all the answers in the back of the textbook with fully opaque post-it notes, and *only* reveal them when you have genuinely tried, and then do corrections. It is OK to fall into the habit of checking full solutions when you are just starting out, but don't come to rely on them, as you can't in an exam. You generally learn more from being wrong. Mark your work as a potentially bored, perhaps tired examiner might - the answers must be accurate and in the correct format. Don't let your efforts slide, day-to-day, or allow yourself to succumb to distractions.

Maths requires alot of brain power, so you need to keep your brain in good condition, if you want optimum results. Exercise regularly, sleep enough, eat well, avoid intoxicants.

Marking your work

After watching the way someone's mental performance was marked by a human during brain surgery on television, and having digested computer errors throughout life, I have come to categorise my wrong answers into the following categories:

When doing practice exam papers, it is a good idea to add up all the wrong marks by the above categories. But they are also useful to use day-to-day, so you can get a rough idea where your issues are, each session. Too many question errors? Perhaps change textbooks. Too many logic errors? Study more before committing to answers. Too many input errors? Slow down, be more meticulous.

Exams

A month or more before exams, it is important to have already covered the course and to be revising all the material that you're familiar with, having iterated through the structured order of a textbook. Note, however, that when you begin revising, your ability to complete questions in the random orders that they are organised will be slow, but with enough practice, will speed up.

Ideally, you want to be working through official exam questions, but given all the changes to the UK exam boards in 2017, there might not be many from your exam board of choice. Suggestions I read online were to be using the 'practice papers' that are found in the back of some textbooks, or to try post-2017 papers from a variety of exam boards.

With CAIE there is an abundance of official past papers publicly, freely available online. CAIE maths papers generally very rarely have errors, so essentially all wrong answers are your own fault. Since 2010, they have released papers to 3 different regions around the world. These papers are usually different, but for less popular papers (e.g. Further Maths, or Mechanics 2), they often duplicate papers among two of the regions, and there is not a highly consistent pattern to how this has been done, over the years, so you have to comb through them to check for duplicates.

University applications

Now, an issue with going it alone is that you may end up with no one you can reasonably request a reference from, who can provide an external assessment of your attitude, work ethic, and progress, when making your application to universities. Furthering, an admissions officer of an oversubscribed university informed me that it is unusual to see people self-studying, upon asking.

The only real solution to this is probably to pay for a tutor, perhaps even only once-a-month, just to check your progress generally. You might prefer not bring them in until you are a good part of the way through the course, e.g. to save money. Perhaps even two tutors, as a backup. I don't honestly have the best solution to this problem, but I would encourage you to investigate, and give this issue serious thought.

Conclusion

After reading this guide, you'll now have a better understanding of the mistakes that are easy to make when going it alone, and understand why I wanted to write this - to help others not make them.

If you have an end goal in mind, such as a course of interest at a desirable university, and you have the time, financial stability, potential, and self-belief, I would thoroughly encourage you to give A-Level maths a try.

[If I had some foresight or planning] I would have taken some deeper math because certainly I've run across things where I have to get help for math.
- Ken Thompson
https://wm-help.net/lib/b/book/2520193410/14

I was interested in math and computers since an early age [...] I was largely self-taught until university.
- Linus Torvalds
https://www.linuxjournal.com/content/25-years-later-interview-linus-torvalds

Everybody is an artist, everybody is a poet. And who told you you weren't a poet? Some square head said [so] at 11, and now you can't draw, now you can't do this. That is what we're against. [But] you are infinite and you have all the possibilities [...]
- John Lennon
https://www.youtube.com/watch?v=mRjjiOV003Q - 25:44