Mathematical tools: a hammer vs a microscope

We congratulate everyone who celebrates the Knowledge Day on 1st September! This post is based on real student stories and illustrates a powerful principle about using appropriate mathematical tools.

A problem in multivariable calculus

The following problem was the first introductory question in a recent exam on multivariable calculus at a top UK university. This question was designed for students who had found the course hard. Indeed you could get a full mark within few seconds using only basic maths from school.

Q1. Find the minimum value of the function $$f(x,y)=x^2+2y^2-4x-4y$$ over all real $$x,y$$.

Derivatives may not be needed at all

Rather surprisingly, only about 2% of the 200+ class remembered the lecturer’s advice to use appropriate tools. The remaining 98% preferred the hard way finding extreme points through partial derivatives. Similarly to 1-variable functions, as we discussed in the post on global extrema, we may often find extreme values of a function without using any derivatives.

Moreover, about 50% of all students claimed that their critical point is a local minimum without justifications, hence lost almost a half of the full mark. The other 48% spent a lot of time justifying that the only critical point (where both 1st order partial derivatives vanish) is indeed a local minimum by using the Hessian of four 2nd order partial derivatives. About 10% of the class actually wrote more than 2 pages on this first question, though only a couple of lines was enough.

Here is the full solution:

$$f(x,y)=x^2+2y^2-4x-4y=(x-2)^2+2(y-1)^2-6\geq -6$$ for all real $$x,y$$.
Hence the minimum value is $$f(2,1)=-6$$.

This simple technique of completing a square illustrates the powerful principle that mastering the basics is more important than trying an advanced method without proper understanding.

Completing a square vs differentiation

Here is the simple mnemonic rule: What can you do with a quadratic polynomial? – Complete a square, of course! Most students usually try to directly write the roots of a quadratic polynomial by using the so-called quadratic formula. Actually, this formula is proved by completing a square, which is a good exercise especially if you have never done it.

From the computational point of view, the quadratic formula is equivalent to completing a square, so we essentially make the same computations in both cases. However, after completing a square, a quadratic polynomial becomes structured (similar to the simplest form $$x^2$$).

Moreover, the same idea of completing a binom $$(x+a)^n$$ works for any higher degree polynomial, but the quadratic formula doesn’t. Actually many real problems in mathematics are about putting various objects (say, functions or matrices or groups) into a normal or structured form.

So completing a square is simple and efficient like a hammer, while differentiation is powerful and delicate like a microscope. A popular student question: “Can I still use a microscope, because I like it?” Our answer: “Yes, you can, though it may look a bit unprofessional.”

• Riddle 19: find the maximum value of $$f(x)=6x^3-x^6$$ over all real $$x$$.
• How to submit: to write your full answer, submit a comment.
• Hint: find a point $$a$$ such that $$f(x)\leq f(a)$$ for any real $$x$$.
• Warning: using derivatives is possible, but is hard to justify.
• Prize: free 1-year access to one of our interactive web tutorials.
• Restriction: only the first correct public answer will be rewarded.
• Update: Carlo has solved the problem, see attempts 1 and 2.

If you wish to receive e-mails about our new posts (for a quicker chance to answer a riddle and win a prize) or distance courses, please contact us and tick the box “keep me updated”. You can easily unsubscribe at any time by e-mailing “unsubscribe” in the subject line.

How to prepare for tough exams: 3 mistakes and 3 tips

1st mistake: starting to worry about exams too late.

One student at a top UK university started to panic about 2 weeks before an exam and e-mailed the lecturer that she is really worried. In our opinion, this worry is a very good sign and 2 weeks are probably the absolute minimum when students should start to worry. Indeed, this student got 83 (of 100) in the exam, while her best mark for any other module was 67. Another student started to e-mail his questions on the course much earlier about 2 months before the exam and then gained 100.

1st tip: regular work wins over last moment rush.

One of the two top revision tips by BBC is distributed practice, namely a continuous training over a long period. Indeed, making many short steps is easier than going up a steep slope. Such a long-term practice is especially important in mathematics, because some concepts often require months or even years to understand properly. That is why we run our distance courses over several months: 12-week courses for the MAT paper and Oxbridge interviews, and 16-week courses for STEP exams.

2nd mistake: reading solutions without trying hard.

Writing rigorously justified solutions to mathematical problems is a time-consuming job even for experts. Most publicly available solutions to past exam questions from MAT or STEP papers are too short and skip important details. If students get solutions for free without making their own efforts, then the learning value is close to 0. On the contrary, if you really tried several approaches and used expert hints, then you are more likely to master key ideas and solve any similar problems.

2nd tip: do problems yourself as in a real exam.

The second (and final) top revision tip by BBC is practice testing, namely doing real (or similar) problems. All other popular tips have either moderate or low value. A long time before these tips were published, we had started our courses by modifying questions from real past exam problems.

Such a modification is especially efficient for checking your understanding. Even changing simple notations does the job. For instance, most students can be confused by a function x(f). Of course, we genuinely modify MAT and STEP problems by changing not only numerical values, but also all forms of given functions keeping essential ideas. Hence reading a past solution can help a little bit, but does not spoil our homework. We often see that students have read a solution to an original past exam question, but have made the same mistakes in proofs that we find in almost every homework.

3rd mistake: paying someone else for your job.

One student surprised us by the following suggestion. He didn’t want to do any STEP-like problems himself, but was keen to pay us money for solving easier problems at A-level. Yes, he was going to train us (and pay us) for A-levels instead of using our guidance for his own exam preparations.

If you are learning to drive, then you will certainly drive yourself under the supervision of an expert driving instructor. So you will not pay for watching how a Formula-1 instructor drives a cheap car, because only watching hardly helps you learn to drive. Similarly in any exam preparations, you may want to do problems yourself and learn from detailed feedback on your regular written attempts.

3rd tip: find an expert to guide your progress.

Some over-ambitious students claim that they can complete our first homework, which we mark for free in any course, but never submit their work. Getting a right answer doesn’t mean a complete solution in mathematics. Actually, the annual STEP examiners’ report highlights that no credit is given for guessing a specific answer without explaining the logic.

So mathematics is not about getting a right answer, but about rigorous justifications how and why you can arrive at a right answer. Even our best students rarely gain the full mark for their first homework. In fact, these students quickly understand that their writing style needs major improvements and start working hard to succeed later.

This time we suggest the following poll instead of a usual riddle.

• Poll 1: what is your best revision technique by your past experience?
• Prize: if you also submit a first correct answer to one of our open riddles,
then you will get a free 1-year access to two of our interactive web tutorials.
• Restriction: only the first vote with a correct public answer will be rewarded.
• Update: Learner has shared excellent advice and also solved our riddle 1.

Some countries are celebrating the Day of Knowledge today as the start of a new academic year, so our warm congratulations to everyone who learns!

If you wish to receive automatic e-mails about our new posts (for a quicker chance to answer a riddle and win a prize) or distance courses, please contact us and tick the box “keep me updated”. You can easily unsubscribe at any time by e-mailing “unsubscribe” in the subject line.

How to choose maths courses: broad range vs narrow focus

Skipping core modules restricts your potential.

We have heard about a recent Oxbridge graduate in maths and philosophy who wanted to apply for a more advanced course at another university with an entrance exam. The entrance exam had 4 problems in basic topics such as calculus and probability. To our surprise, the student did not study any probability and decided not to apply. Yes, the probability theory is a key tool of modern mathematics and should be studied by all undergraduates who want to use their mathematical ability in the future.

Why are there so many optional modules?

The same student eagerly argued that their university offered the widest possible choice of maths modules in the UK including ‘Axiomatic Set Theory’, ‘Analytic Topology’, and ‘Banach Spaces’. To avoid an inevitable disappointment, the student wasn’t told that all these topics are actually compulsory for any maths undergraduates at many overseas universities. The main reason for creating optional modules is the tuition fees and the consumer culture at UK universities. Compulsory modules can make some students unhappy, which is considered unacceptable when the top priority is the student satisfaction, not the learning outcomes.

How a wide choice creates wider problems.

Many graduates with the same 1st class degree from the same maths programme at the same top university often have completely different knowledge and skills. Hence employers can not quickly and fairly select best candidates and need to further analyze all examined modules (and their content) or simply design their own entrance exams or job interview tests.

Similarly, all top UK universities require proper entrance exams such as STEP and MAT, which seem better for choosing brightest mathematics candidates than an inevitable random selection from a large pool of school students with the highest grades A*A*A* at A-level.

How can you benefit from flexibility?

One of our students couldn’t get required STEP grades for a maths programme at Cambridge. However he was flexible enough, talked to admissions tutors about other options and got in Cambridge on a joint programme with computer science. Indeed, many universities allow students to change their programme later, so a purely mathematical course at Cambridge was still a option. Moreover, a solid foundation in two closely related subjects such as mathematics and computer science is simply wider, hence more robust for a future career.

Returning to our first story about the Oxbridge graduate, yes, we think that any computer science student should learn the probability theory, at least because randomized algorithms are now a large and active research area in computer science.

Another story is about a student from Strasbourg, who was on an exchange programme for a year at a top UK university. The student enrolled on 7 modules instead of required 6 despite clashes in the timetable between lectures. The student gained 90+ marks in all 7 exams and came on top of all her classmates.

What to learn: broad range vs narrow focus.

In the past all great inventors were universal specialists in many subjects. For example, was Leonardo da Vinci a painter, sculptor, architect, musician, mathematician, engineer, writer, anatomist or “all in one”? Actually, the word “university” hints at universal knowledge.

Distant research areas converge to each other and become inter-disciplinary. For instance, cancer genetics uses fast algorithms from computational geometry, which are rigorously justified by methods from (applied?) statistics and (pure?) topology. Analogous interactions of distant subjects in the computer vision have led to robotic cars that you will be able to buy soon.

It seems that the world of the (near) future will need only super-specialists. If you play chess, you hopefully know standard positions when a king can keep 2 perpendicular aims in mind by making diagonal moves, which are simply longer. Similarly, by learning topics on the interface of different areas, you can keep your options open and make a right choice later when you have more information. We finish by the following riddle with our usual reward.

• Riddle 4: find the product of the roots of of the equation $$0.5x^2+52=99x$$.
• How to submit: to write your full answer, simply submit a comment.
• Hint: there is no need to find the roots since only their product is needed.
• Warning: please avoid using a calculator, because you can be smarter.
• Restriction: only the first fully correct public answer will be rewarded.
• Prize: free 1-year access to one of our interactive web tutorials.
• Update: after attempts 1 and 2 Ariella has won the prize.

If you wish to receive automatic e-mails about our new posts (for a quicker chance to answer a riddle and win a prize) or distance courses, please contact us and tick the box “keep me updated”. You can easily unsubscribe at any time by e-mailing “unsubscribe” in the subject line.

How to enter top UK universities: 3 mistakes and 3 tips

We share a few anonymous stories how students succeeded or failed to enter top UK universities for maths degrees. We assume that our readers are familiar with our summary about top UK universities and maths entrance exams. This picture is only a rough guide:

Mistake 1: a late start in exam preparations.

Surprisingly many students ask us to help just about a week before their exams. It is good that these students try hard to submit as many homework problems from our distance courses as they can. It turns out that some of these late students are actually rather strong in mathematics. However, a week or a month seems insufficient, simply because learning deep maths concepts often takes months and even years. Many late students unfortunately fail.

Tip 1: try a similar real exam in advance.

The top UK universities for maths degrees (Cambridge, Oxford, Warwick, Imperial) usually give their candidates only one chance to gain a good grade at their entrance exams. For instance, Oxford selects candidates for interview in December by their results in a MAT paper that can be taken only in November.

We taught a strong Oxford candidate who couldn’t complete MAT for personal reasons that emerged on the exam day. However, the student gained a grade 1 in the harder STEP I exam earlier in June. So Oxford admissions tutors happily accepted this grade 1 in STEP I instead of MAT and invited the student to the interview. The student successfully entered Oxford.

Mistake 2: are personal statements important?

Another our student was preparing for a MAT paper, but submitted only few first homework problems from our course. About 2 weeks before the exam, the student e-mailed us their personal statement and asked for our opinion. The statement seemed brilliant, though we are not experts in assessing personal statements. However, by our past experience, a better progress in our course was needed to succeed in MAT. So we encouraged the student to focus on their MAT preparations. Unfortunately, it was a bit too late in this case.

Tip 2: focus on maths exam preparations.

University admissions tutors are often lecturers or professors who are pretty busy with their research and teaching. Moreover, mathematicians actually prefer an objective selection that is based on already gained results in real challenges. Shortly, you could outline your achievements (as a bullet point list) in maths competitions such as challenges by the UK Mathematics Trust. It seems worth spending much more time (95% or 99%) on training for proper challenges and entrance exams that bring long-lasting rewards.

Our students tell us about (and often thank us for) their success in the British Mathematics Olympiad, Senior Team Challenges and Senior Kangaroo Challenges despite we run distance courses only for entrance exams such as STEP, MAT papers and Oxbridge interviews. Smart students consider any serious olympiads or challenges as extra opportunities to practice problem-solving skills in non-standard situations beyond the school curriculum.

Mistake 3: over-estimating your own potential.

A few years ago one year 12 student gained a diploma in the maths competition at our summer school for maths candidates to top universities. Later the student applied to Oxford and decided to independently prepare for a MAT paper without taking our first distance course. Unfortunately, the application was unsuccessful. Then the student took a gap year, applied to Cambridge and completed our 4 courses for Oxbridge interviews and 3 STEP exams. This second attempt was better: a grade S in both STEP II and III, and a place at Cambridge.

Tip 3: learn from past mistakes of others.

Many people learn much more from mistakes than from successes (if it’s not fatally late). Watch this brilliant show by one over-optimistic hopeful who didn’t properly prepare for a challenge. If you would like to feel better after exams, hopefully you know what to do now.