Master Maths

Study tips for maths candidates to top universities

What is mathematics? And what is it not? Master maths: what does it really mean? Writing style at maths exams: in English! Timing for problem-solving at a real exam Exercise physically to excel in mathematics

Study diary: self-organisation for exams Revision guide: work and revise regularly How to self-learn on distance courses What meals to consume in our courses Step-by-step tips from a STEP student

We e-mail more (individual) study tips to our students during each course, contact us to enrol

What is mathematics? And what is it not?

• Not about recipes to get numerical answers without explanations.
• Maths is about logical arguments that justify all steps and results.
• About rigorous proofs that convince anyone including experts.
• About powerful methods that can work in many general cases.
• The main question is not "what?", but "why?" and "why not?"
• Mathematics is hard, hence mathematicians are highly valued.

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Master maths: what does it mean? You will

• recognise patterns: when to use specific methods or results
• choose a best strategy: a shortest path to a correct solution
• write rigorous proofs and make sure that answers are correct
• link maths concepts together to develop mathematical thinking
• check if you have really understood a mathematical concept or a proof:
  only if you can explain it to another person whose level is below yours
• joke: "I was explaining to my class that two halves are always equal and
  have finally understood it myself, but the larger half of the class hasn't got it".

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Writing style at maths exams: in English!

• STEP examiners' reports say that: "it comes the need for explanations in English ...
  equations with no explicit connections between them can leave the reader confused".
• If you are heading to an English university, then your maths solutions should contain
  at least basic English: "because", "if...then", "hence", "since", "assume", "we prove".
• How to improve your writing style? More words, fewer symbols and practice!
• Imagine: you try to convince someone of your age from another school.
  Then simply write all words and logical arguments that you pronounced.
• Your aim is to convince a marker that you have mastered the subject.
  Concise justifications in maths are the art requiring years of practice.
• Use all our detailed step-by-step solutions as a model for your own.
  Pay attention to our comments and learn to write in a similar style.
• Little credit is given for answers without enough justifications in STEP.
  A full solution to STEP problem often takes at least 2 hand-written pages.
• Students who enjoy learning from their mistakes during a proper training
  become confident that they will avoid the same mistakes at a real exam.

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Timing for problem-solving at a real exam

• The Mathematics Admissions Test (MAT) is 2.5 hours long, hence
  the aim is to spend 5 min on each of 10 multiple choice questions
  where only numerical answers are required and about 25 min on
  each of 4 long problems where logical arguments are assessed.
• Each STEP paper is 3 hours long and only 6 solutions are counted,
  so the aim is to spend 30-45 min on 4-6 problems easiest for you.
• As in any training, the accuracy is more important than the speed,
  hence the initial aim is to write fully justified solutions learning
  from our feedback and using step-by-step hints if necessary.
• Know your strengths (from your study diary): in STEP your best 6
  solutions are counted, start from problems that are easiest for you.

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Exercise physically to excel in mathematics

• Physical activity reorganises the brain to be more resilient to stress.
  If you can run 3 miles, then you can sit a 3-hour exam without fatigue.
• A physically fit candidate with enough willpower to exercise regularly
  is more likely to succeed in any maths competition than a couch potato.
• Our tutors at a younger age and many your international rivals right now
  could run 12km in 1 hour, ski 75km or cycle 150km in a day (with breaks).
• We can work hard by providing quick and detailed feedback to our students,
  because even now our tutors run 5.5-6 km for 28-30 min 3-4 times a week.
• Many top mathematicians are active in sport and enjoy a good physical form,
  e.g. Andrey Kolmogorov and many his students are now famous for making
  mathematical breakthroughs during their kayaking trips and mountain hikes.
• We recommend physical activities two-three times a week for 25-30 min
  up to 65-85% of your maximum heart rate (roughly: 220 - your age).
  Of course, consult your doctor if you have any concern or condition.

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Study diary: self-organisation for exams

• start a few months in advance as advised by BBC Future
• note your progress on each topic or problem in a study diary
• how long did it take for you to write a complete solution?
• what hints (general hints or step-by-step hints) were used?
• students may attach our detailed feedback on their script
• before exams: revise topics where your progress was weaker
• the diary is personal, not for us, not for teachers or parents.

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Revision guide: work regularly and revise often

• Top tips by BBC: revision techniques - the good, the OK, the useless
  practice testing: we mark every homework problem as a mini-mock,
  distributed practice: we flexibly run our courses over a few months.
• Work at least two-three times every week with frequent breaks:
  25 min study + 10 min break + 25 min study + 10 min break +...
• Submit weekly written homework and learn from our feedback,
  use our final tips and avoid common mistakes in each problem.
• Revise the material from the previous weeks or topics, because
  repetitions naturally strengthen neural connections in your brain.
• After a lesson, your brain forgets more facts in the next 8 hours
  than in the subsequent 30 days, regularly revise to retain facts.
• Learning in a hurry before an exam damages your performance,
  sleep well (not only before an exam) to keep your brain fresh.

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How to self-learn on distance courses.

• Read an outline of each topic and our introduction before a problem.
• Independently attempt every problem without hints for at least 5-10 min.
• If needed, read our general hints and then attempt the problem try again,
  then read our step-by-step hints and try hard to write your own solution.
• E-mail us if you are a stuck at some step, we are always happy to help.
• Finally study our solution, common mistakes of students and final tips.
• Similarly approach your homework, always submit even if it is late.
• If you independently completed homework, read our hints just in case
  to make sure that you haven't missed anything and has justified all steps.
• If our comments on your script seem unclear, e-mail and we shall clarify.

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What meals to consume in our courses

• After enrolling, focus on the key meals of our complete courses, namely
  learning from our detailed feedback on homework and our e-mail support.
• Students who only read notes without submitting their homework consume
  only a portion of salad, but will compete with other properly fed candidates.
• Regularly take our innovative supplement: interactive web tutorials give
  step-by-step hints and allow multiple attempts to boost your confidence.
• Best students follow a healthy diet beyond externally provided meals and
  always attempt all training problems (with hints if needed) from our notes.
• If you aim to win a real-life marathon like STEP or an Oxbridge interview,
  keep going and complete a full course to build enough strength and stamina.
• The STEP examiners' reports every year highlight that "... it tests motivation...
  Those who are not willing to make the effort are unlikely to thrive on a course".

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Step-by-step tips from our STEP student

• I copied out the notes on each topic (our comment:
  writing down maths formulae may help remember them).
• Did the training problem, then after comparing with the solution,
  wrote out a 'model solution' with some of my original working, and
  some of the solution if I thought that was a better/more concise way!
• Did the homework problem (our comment: and review our feedback,
  if there are any questions, we are always happy to advise by e-mail).
• Copied out the common mistakes & useful tips for each problem (our comment:
  common mistakes and tips are regularly updated after almost every homework).

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