Author Archives: Master Maths blogger

Success in Oxbridge interviews: 3 mistakes and 3 tips

training-test-vs-life We wish good luck to all Oxbridge candidates at their interviews in December!

Mistake 1 : are numerical answers important?

Numerical answers have very little value in mathematics and are even less important in real life. Indeed, computing the page rank of a single web page is probably worth about 1 pence. However, the page rank algorithm (with many others) can quickly process the whole World Wide Web, which turned Google into a multi-billion company.

At a proper mathematics interview you are expected to demonstrate your thinking process. If interviewers are interested only in your numerical answers, they can get them in a much quicker and cheaper way via an on-line test.

Tip 1 : never believe and learn how to prove rigorously

A few years ago one student argued that he should have got a full mark for his solution that gave a correct numerical answer justified only by the following phrase: “I believe that my answer is correct”. Students who are interested in beliefs will probably be welcomed in theology. Mathematicians are always expected to prove new results, for instance by using previously proved theorems.

Mistake 2 : the easy ways to make a bad impression are

  • being nervous, e.g. if you didn’t sleep well before
  • talking too quiet or looking physically weak or tired
  • reflecting on your performance during the interview
  • asking inappropriate questions, e.g. how have I done?

Oxbridge interviews are usually in the morning, so you need to have a regular sleeping pattern at least a few weeks before your interview, not only at the very last night. Read our advice on physical exercise. There is no time and no point to think or to ask if you have done well before all interviews are finished.

Tip 2 : read interview stories from past students

We have collected a few useful stories from our past successful students about their Oxbridge interviews and share this first hand experience with all our current students. For instance, all Cambridge candidates and some Oxford candidates sit a written test before oral interviews.

Some of our students happily announce shortly after their interviews that they have done well, e-mail us their questions and later receive rejections. In most these cases we clearly see that the questions were quite easy. If you get easy questions, you will not be told that you are considered as a weak candidate, because the job of interviewers is to keep all candidates happy.

One of our best students last year e-mailed that he could hardly complete any question and only with a lot of hints from his interviewers. However his questions were much harder than from other students. Actually, he received a standard Cambridge offer with a grade 1 in both STEP II and III, then gained two grades S after completing our STEP courses.

Mistake 3 : avoiding proper feedback on your progress

We regularly encounter over-optimistic candidates who are self-studying without any expert advice on their progress. Then the first (and often last) feedback will be a “yes/no” from university admissions. The MAT examiners and Oxbridge admissions tutors guard hard all exam scores and STEP candidates can hope to receive only their numerical mark. So the current entrance exams at top UK universities provide little feedback to students after months of self-studies.

The education systems outside Europe and North America are quite opposite. Almost all exams are oral and students naturally get a lot of feedback. Oral exams are often harder and harsher, so it is a proper training for real life. That is why the key value of our distance courses is the detailed feedback on regular homework, where questions are usually harder than in past exams.

Tip 3 : don’t train for a test, but aim higher

A pupil told a kung-fu master: “I have been training really hard for many months, but I still can not break through that board”. The master watched his attempt and then said: “If you hit the board, you will never break through it. You should hit beyond the board.”

Similarly in any learning, if students are trained only for a specific test, they are likely to fail, read
BBC Education: most A level grade predictions wrong. The winners always aim higher (much higher than their target) and that is why they often win even if something goes wrong.

The usual feedback from our students on the distance course for Oxbridge interviews: “your Oxbridge questions are much harder than anything I have done before at school”. Here is our powerful learning principle: the harder the training, the easier the exam!

  • Riddle 9: Find all real solutions of the inequality \(x\geq\sqrt{3x-2}\).
  • How to submit: to write your full answer, submit a comment.
  • Hint: remember that \(\sqrt{x^2}\neq x\), read answers to this riddle.
  • Warning: the very first step in most common attempts is wrong.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Restriction: only the first correct public answer will be rewarded.
  • Update: Paul solved the riddle and won a prize, read our comment.

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What is important for success: building vs enthusiasm

building-vs-enthusiasm Last time we quoted a report from BBC. This time we shall give our personal perspective on a report by Sky News: Wales School To Help World’s Poorest Children.

Students can learn under a highway

The Sky News report shows a class in India of about 80 primary school pupils who are sitting on the ground under a highway bridge and without walls. Other pupils from a very different primary school in Wales were so impressed that they started a campaign to collect donations across entire Wales for the Indian school.

Some students may struggle without a building

About 2-3 years ago there were a lot of stories about cancelled building projects for schools because of austerity measures. Namely, the BSF programme (Building Schools for the Future) was cut. We are actually wondering if future schools will need any buildings at all.

Other students may learn without sunlight

We know one Physical-Mathematical school in a 1-million city somewhere near the geographic border between Europe and Asia that didn’t have any building in their first year of existence.

This school was set up like the so-called “free schools” in the UK. However, there was the essential difference: new students were accepted only through proper written exams in mathematics and physics, while the free UK schools are “free from” (can not use) any academic barriers.

As it often happens, the building wasn’t ready for a new academic year and the teachers rented a few rooms in a nearby ordinary school so that a half of students had to start their lessons in the afternoon in the so-called second shift. The most advanced class of only 10 students was lucky due to the small size and always occupied a room of about 2.5 by 5 metres with a window. Only 6 double tables were crammed into this “classroom”, which was also a through-passage to another room regularly used by the host school teachers right in the middle of lessons.

Developing resilience is a key to success

Four of these 10 students gained prizes in the regional olympiads in mathematics and physics (population size about 3 million as in Wales). Later 2 students received 3rd diplomas in the final stages of the national Maths Olympiad and the Soros Maths Olympiad (sponsored by George Soros) in the country whose population is twice the UK size. From this class of 10, one former student is now a mathematician in the UK, another one is a physicist in the US, one more is a software engineer in Microsoft headquarters and at least 3 more are computer programmers.

Other larger classes were less lucky and were put in a basement without windows. If you think that teaching in a basement contradicts health and safety regulations, we could say that the words like “regulations” and “policy” may mean corruption and bribes in many countries outside Europe and North America. Namely, if bureaucrats come to check health and safety, they come simply for money, not even for ticking boxes let alone health and safety.

Once after school lessons, the oldest students (yes boys, not girls) were asked to help lift bricks to the upper floor of their new school under construction, simply because there were too few available builders who couldn’t cope. So nobody cared about health and safety when future PhDs worked for free with bare hands. This experience was actually very positive, the boys really enjoyed “building” their school, though the building was finished only after their graduation.

Enthusiastic teachers can make a difference!

The Sky News report about the Indian school without walls correctly highlighted the enthusiastic teacher who manages to teach the class of 80 students. The Physical-Mathematical school we mentioned above also had enthusiastic teachers who simply moved from another “Physical-Mathematical” school where the head teacher used all extra money (given to specialised schools) for teaching economics, not maths.

You might think that these extra money could have been spent for science visits or for inviting maths experts or for running olympiads and outreach activities? No! The money were used simply for feeding the class of 10 students who won all possible maths olympiads up to the regional level and sometimes higher.

The current so-called “austerity times” in the UK have nothing common with a real collapse of a failed state. Here is a quote from a famous mathematician who escaped to Paris at that time: “French mathematicians work to eat well, Russian mathematicians eat to work well”. It was a really happy time when the class was fed twice during the day and went home ready for doing more maths without thinking about food at least until late evening.

Only 15 years later the same school teachers and their new students had more time and money for proper teaching and learning, not only for feeding students. Indeed, the students from this “far away school in the middle of nowhere” excelled at a much higher level having won

  • two gold medals at the International Physics Olympiads
  • two silver medals at the all-Chinese Maths Olympiad
  • a silver medal at the International Biology Olympiad
  • a gold medal at the International Junior Science Olympiad.

As a final remark about real-life challenges, here is the announcement at the school website from December 2012: “the olympiad training of the city team on Saturday is cancelled because of a water cut in the school”. Health and safety? However, the enthusiasm always wins!

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Next Big Thing in online education: SPOCs vs MOOCs

SPOCs-vs-MOOCs+

BBC News has reported the new term SPOC in Next Big Thing. We shall remind the short story.

Distant past: bricks-and-mortar universities.

For hundreds of years students learned in so-called bricks-and-mortar universities. The general scheme includes
(1) paying large tuition fees for
(2) access to real professors and also
(3) enjoying social life away from parents
(4) hopefully gaining a degree at the end.

However the Internet is changing the education.

Recent past: MOOCs (Massive Open Online Courses).

Many students have probably heard the term MOOC, see more details in the Wikipedia article. The term MOOC was coined in 2008 and first 3 real MOOCs appeared in September 2011. By September 2013 there were dozens of MOOC providers in different countries. Initially MOOCs didn’t have the same factors as the conventional universities:
(1) large tuition fees
(2) real professors
(3) any social life
(4) certificates.

Several MOOCs have started to offer
(1) discussions and chats with instructors
(2) assessments with identities verified
(3) a small charge for certificates.

The main drawback of MOOCs has been the low retention rate: about 8% of all enrolled students actually pass a final test.

Next Big Thing: SPOCs (Small Private Online Courses).

It seems that term SPOC has just been coined by BBC on 24th September 2013. There is no Wikipedia article on SPOCs yet: checked on 27th September 2013. So we shall briefly describe the key points of the BBC report.
(1) Harvard says that we are already in the “post-MOOC” era.
(2) Access to SPOCs will be restricted to dozens or hundreds.
(3) Proper guidance will be provided for the selected students.
(4) Students will be more rigorously assessed than in MOOCs.

We have started our SPOCs (distance courses for maths candidates to top UK universities) in September 2010, which was 1 year before any MOOCs appeared and 3 years before Harvard started to think about SPOCs. We hope that other SPOCs (will) have the same key values as our current courses for top students preparing for MAT papers, Oxbridge interviews, STEP exams:
(1) quick and detailed feedback on regular written homework
(2) training on many problems harder than past exam questions
(3) best tips and advice from our recent successful students
(4) interactive quizzes offering gradual hints when needed.

Actually, we feel rather happy that big players like Harvard are catching up. The flexibility of small start-ups is a great advantage in tech innovations.

Small mammals are now ruling the world, while huge dinosaurs were wiped out. Here is the powerful principle: the size does not matter, but the skills do!

  • Riddle 7: which black area is larger in the picture in the top left corner
    at the very beginning of this post: all small squares or two big squares?
  • How to submit: to write your full answer, submit a comment.
  • Hint: you may assume that each small black square has side 1.
  • Warning: please mathematically justify your conclusion.
  • Restriction: only the first correct answer will be rewarded.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Update: after Ariella’s attempt, the answer has been explained.

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Mistakes in sketching graphs: easy, common, unexpected

coordinate-axes-2directions Easy mistake: an axis can’t have 2 directions.

Several our students persistently started to sketch graphs from the diagram above. By definition, an axis is an infinite straight line with one direction, so we need to know a direction of increase of the variable along the axis. In computer programming the axes have different directions, because we usually type on a screen from top to bottom starting from the top left corner (the origin). coordinate-axes-graphics
So the vertical \(y\)-axis is oriented downwards and the parabola \(y=(x-8)^2\) has the reversed shape on such a diagram. However, if two directions are specified on the line of the \(y\)-coordinate, then we can not know how to draw even the simple parabola \(y=(x-8)^2\). By the standard convention in mathematics the horizontal \(x\)-axis is oriented to the right hand side and the vertical \(y\)-axis is oriented upwards. coordinate-axes-standard

Common mistake: corners/cusps vs smooth points.

Many students often draw the graph \(y=2\sin^2 x\) with corners and cusps at the points
\(x=\frac{\pi}{2}+\pi n\) for any integer n, where the graph meets the \(x\)-axis. Starting from the familiar graph \(y=\sin x\) and before scaling it by factor 2, the students probably reflect the negative parts to the upper half-plane and get the following picture with cusps (“acute corners”). graph-2sinx-squared-cusps This reflection should give right-angled (not acute) corners, because \(y=\sin x\) meets the \(x\)-axis at angles \(\pm\frac{\pi}{4}\) since the gradient is \(y'(x)=\cos x=\pm 1\) at \(x=\frac{\pi}{2}+\pi n\).

However, \(y=2\sin^2 x\) has neither corners nor cusps, because the derivative \(y'(x)=2\sin x \cos x =\sin 2x\) is well-defined everywhere. Indeed, \(y'(x)=sin 2x=0\) at \(x=\frac{\pi}{2}+\pi n\), so the graph the \(x\)-axis smoothly touches \(y=2\sin^2 x\) at all the points \(x=\frac{\pi}{2}+\pi n\) as in the correct picture below.

graph-2sinx-squared-smooth

Actually, best students know the formula \(2\sin^2 x=1-\cos 2x\) and sketch the simpler graphs \(y=\cos x, y=\cos 2x, y=1-\cos 2x\).

Unexpected mistake: Q3(v) in the MAT paper 2011.

Q3 in the MAT paper 2011 is about the cubic parabola \(y(x)=x^3-x\) and its tangent line \(y=m(x-a)\) having a slope \(m>0\) and meeting the \(x\)-axis at a point \(x=a\leq 1\). cubic-parabola-meets-tangent

Through any point (a,0) for \(a<-1\) we can draw 3 tangent lines to \(y=x^3-x\). One the them touches the cubic parabola at a point \(0<x<1\) and has a negative slope. However, there are two different tangent lines with a positive slope. Only one of them was shown in the original problem as in the picture above. No restrictions apart from \(m>0, a\leq -1\) were given in the problem. Hence we may have the second tangent line for \(b<a<-1\) in our picture below. cubic-parabola-unbounded-area

Part (ii) asks to prove that \(a=\frac{2b^3}{3b^2-1}\). This formula also works for both tangent lines: if \(b=-2\), then \(a=-\frac{16}{11}>b=-2\).

Part (iii) asks to find an approximate value of b when \(a=-10^6\). The examiners’ solution of part (iii) considers only one possibility when \(a=\frac{2b^3}{3b^2-1}\) has a large (absolute) value because of a small denominator, which implies that \(b\approx -\frac{1}{\sqrt{3}}\) (if negative). However, the second possibility is that b also has a large (absolute) value. For instance, if we set \(b=\frac{3}{2}a\), then \(\frac{2b^3}{3b^2-1}=\frac{\frac{27}{4}a^3}{\frac{27}{4}a^2-1}=a+\frac{a}{\frac{27}{4}a^2-1}\) is approximately equal to \(a\) when \(a\) has a large absolute value. So another tangent line at \(b\approx -\frac{3}{2}10^6\) meets the x-axis at the same point \(a=-10^6\).

Part (iv) asks to show that the tangent line meets the cubic parabola at the second point \(c=-2b\). This formula also works for both tangent lines: if \(b=-2\), then the tangent line \(y=11\Big(x+\frac{16}{11}\Big)=11x+16\) meets \(y(x)=x^3-x\) at \(x=c=4\) where \(y(4)=60=11\cdot 4+16\).

Part (v) asks to find the largest possible area of the region R bounded above by the tangent line and bounded below by \(y=x^3-x\). For both tangent lines, the region R is finite because both
tangent lines eventually intersect the cubic parabola at \(c=-2b\). The examiners’ solution to part (v) says: “We can see that as \(a\) increases then the tangent line rises and so the area of R increases. So the area is greatest when \(a=-1\)“.

This claim holds only for the first tangent line when \(-1\leq b<0\). However, our last picture with the second tangent line above shows a much larger area when \(b<-1\). Actually both tangent lines coincide when \(a=-1\), so one family of tangent lines for \(a<-1<b<0\) is joining another family of tangent lines for \(b<a<-1\). The picture in the problem with only one tangent line for \(a<b\) degenerates in the case \(a=b=-1\) when two points A,B merge and can hardly be used as a reference for part (v).

It is rather surprising that Q3(v) in the MAT paper 2011 and its solution have not been amended in Oxford MAT 2011 solutions and in Imperial MAT 2011 for almost 2 years after the actual exam: checked on 20th September 2013. You can find more details on this problem in our web tutorial Tangent lines and areas bounded by cubic parabolas.

Mathematics is a wonderful subject, because students can check all arguments and find more efficient solutions themselves. That is why we keep learning from our smart students.

  • Riddle 6: does \(x^2+y^2=1\) define a function \(y(x)\) for \(-1\leq x\leq 1\)?
  • How to submit: to write your full answer, simply submit a comment.
  • Hint: sketch the curve \(x^2+y^2=1\) on the plane, you may try to express \(y(x)\).
  • Warning: a function \(y(x)\) should have a single value of \(y\) over \(-1\leq x\leq 1\).
  • Restriction: only the first fully correct public answer will be rewarded.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Update: Carlo has solved the problem, see the comment.

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How to properly state mathematical theorems: 3 key parts

2sided-bridge-vs-1sided
At summer schools we often asked to state Pythagoras’ theorem. The most popular answer was short: \(a^2+b^2=c^2\). Initially we were rather shocked that many students haven’t probably seen any properly stated mathematical theorems.

Now we are more experienced and will give more hints. The first light hint: could you write Pythagoras’ theorem in more details, e.g. by using words? Here is the next popular attempt (literally!): “a squared plus b squared equals c squared”.

Then we ask: what do you mean by \(a^2+b^2=c^2\)? Is it an equation you are going to solve or what? Let us give a parallel example from history.
Question: who is the Queen of England?
Answer: the second!
The numerical answer is correct, but can you expect a full mark? Similarly in mathematics, the equation \(a^2+b^2=c^2\) says very little about Pythagoras’ theorem.

At this moment most students draw a triangle with sides a,b,c. With some extra help the triangle is marked as right-angled, not “right triangle”, which is another common confusion. So the statement now says: “A right-angled triangle with sides a,b,c satisfies \(a^2+b^2=c^2\)“. Almost correct: Pythagoras’ theorem is about any right-angled triangle, not just “a triangle”.

There is a slightly longer, but more logical statement: if a triangle with sides \(a\leq b<c\) is right-angled, then \(a^2+b^2=c^2\). Please notice that we have clearly separated the geometric condition (a triangle is right-angled) and the algebraic conclusion (\(a^2+b^2=c^2\) holds).

We continue: great, this is the correct first half of Pythagoras’ theorem, could you state the second half? Even good students who have easily written the first half are often stuck and admit that they don’t know the second half. Another hint: is the triangle with sides 3,4,5 right-angled? Everyone says “yes”, so we ask why. The popular answer: by Pythagoras’ theorem.

Then we repeat: do you really claim that our statement above implies that the triangle with sides 3,4,5 is right-angled? Most students are confused, but sometimes we receive this answer: “it’s just the opposite”. Opposite sides of most coins are different and so are opposite statements in mathematics.

It is easier to reverse the longer logical statement: “if \(a,b,c>0\) satisfy \(a^2+b^2=c^2\), then the triangle with the sides a,b,c is right-angled.” Notice how we swapped the two key parts of the statement: if the algebraic condition is given, then the geometric conclusion holds.

Now we ask to write two statements together as full Pythagoras’ theorem. A popular attempt is to write one phrase after another: “Any right-angled triangle with sides a,b,c satisfies \(a^2+b^2=c^2\). If \(a,b,c>0\) satisfy \(a^2+b^2=c^2\), then the triangle with the sides a,b,c is right-angled.” We clarify: can you write a shorter single phrase by making a logical connection between the algebraic and geometric parts?

After talking about “if… then“, “only if” or “only when“, we arrive at the following statement of full Pythagoras’ theorem: “Numbers \(a,b,c>0\) satisfy \(a^2+b^2=c^2\) if and only if the triangle with the sides a,b,c is right-angled.” The most important part is the keywords “if and only if” saying that the theorem works in both directions.

So any mathematical statement consists of 3 key parts:

  • a given condition (that is given and can be used later in a proof)
  • a logical connection (between the condition and the conclusion)
  • a resulting conclusion (that should be deduced from the condition).

Many statements are one-sided: if A then B (if a condition A is given, then a conclusion B holds). These statements play the role of a one-side bridge between villages A and B. Namely, villagers can go from A to B, but not from B to A.

All great theorems in mathematics are two-sided: A if and only if B, which establishes a two-sided bridge between A and B, so villagers can freely travel between A and B. For instance, Pythagoras’ theorem is a two-sided bridge between geometry and algebra.

Similar and more general statements led to the development of analytic geometry where we rephrase a geometric problem in algebraic terms, solve by using algebraic methods and then translate our solution back to geometry.

You have hopefully appreciated the fascinating fact that triples of positive numbers satisfying \(a^2+b^2=c^2\) are equivalent to geometric shapes of right-angled triangles. Simple Pythagoras’ theorem stated above can be generalised to many (even infinite) dimensions, also works for functions instead of segments and makes sense in very general so-called Hilbert spaces.

In all these cases Pythagoras’ theorem illustrates the powerful principle: 2-sided bridges are much better than 1-sided ones.

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

short-steps-vs-steep-slope 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?
  • How to submit your vote: to write your answer, submit a comment.
  • 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!

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How to choose maths courses: broad range vs narrow focus

broad-range-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.

Is it really possible to learn more? Yes!

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 solve hard problems: a key principle with examples

from-easy-steps-to-hard-problems  It would be much nicer to have a simple problem instead of a hard one. So a natural way to solve a hard mathematical problem is to make it simpler.

The key principle is to simplify.

At summer schools we often ask the simple question: how will you solve the equation \((x-2)^2=1\)? Most students expand the brackets and then apply the quadratic formula.

The standard quadratic formula is a correct general recipe. However, a simple computer program will easily beat you. While you are solving one quadratic equation, the chip in your mobile phone can solve a billion of similar equations. So mathematics is not a cookbook of recipes, but a beautiful subject where efficiency, not brute force, is highly valued.

Efficiency often wins over brute force.

If we replace \((x-2)^2=1\) by a very similar equation \((x^2-2x)^2=1\), then expanding the brackets will produce a longer and more complicated equation of degree 4. Moreover, any computer program based on the standard quadratic formula will fail.

However, both equations \((x-2)^2=1\) and \((x^2-2x)^2=1\) have the same pattern: a square equals a number. Hence the equations can be solved in the same way. For instance, we could replace the expression under the square by a new variable.

A substitution is a common way to simplify.

In the equation \((x-2)^2=1\) we could set \(y=x-2\). Then we get the simpler equation \(y^2=1\), which has two roots \(y=\pm 1\). Hence the original equation \((x-2)^2=1\) has two roots \(x=y+2=-1+2=1\) and \(x=y+2=1+2=3\).

Similarly, in the equation \((x^2-2x)^2=1\) we could set \(y=x^2-2x\). Then we get the simpler equation \(y^2=1\), \(y=\pm 1\). Hence the original equation \((x^2-2x)^2=1\) is equivalent to the union of 2 simpler quadratic equations \(x^2-2x=-1\) and \(x^2-2x=1\).

How to simplify: try to keep a pattern.

Most students would probably solve the last two equations by using the standard quadratic formula again. We could actually continue the simplification and complete a simple square instead. Namely, \(x^2-2x=-1\), \(x^2-2x+1=0\), \((x-1)^2=0\), so \(x=1\) is a double root. Similarly, \(x^2-2x=1\), \(x^2-2x+1=2\), \((x-1)^2=2\), \(x-1=\pm\sqrt{2}\), \(x=1\pm\sqrt{2}\).

There are many more complicated equations with the same pattern \(y^2=1\). For instance, try to solve the equations \((x^3-2x-1)^2=4\),   \((\sin x+\cos x)^2=1\),   \((e^{2x}-2e^x)^2=1\).

  • Riddle 3: simplify \(\sqrt{x^2}\) for any real number x and explain your conclusion.
  • How to submit: to write your full answer, simply submit a comment.
  • Hint: the square root \(\sqrt{x}\) of any positive real number \(x\) is always positive.
  • Warning: many 2nd year maths students at a top university get it wrong.
  • Restriction: only the first fully correct public answer will be rewarded.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Update: R S has solved the riddle, read our confirmation.

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:

uni-entrance-exams

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.

If you would like to share your useful experience with maths entrance exams, please leave a comment by clicking on “Leave a reply” above or e-mail Master Maths blogger.

  • Riddle 2: find the largest number consisting of only 3 symbols: 9, 9, 9.
  • How to submit: to write your full answer, simply submit a comment.
  • Hint: justify that your answer is the largest number among few alternatives.
  • Warning: maths isn’t about recipes, but requires thinking “outside the box”.
  • 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, the answer has been explained.

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.

Introduction to our Master Maths blog: 4 reasons to start

Master Maths logoWe have the following 4 reasons to start our Master Maths blog:

  • answer popular questions from our mathematics students
  • share our very different educational experience in maths
  • share learning experience of our students (anonymously)
  • provide a forum for exchanging and generating new ideas.

We assume that our readers are smart and can make their own conclusions from our stories, so we shall try to be open-minded and give real-life examples instead of unjustified advice.

In our blog we shall give regular tips and advice how to

  • concisely write full mathematical statements, see our first riddle below
  • use proper notations in solutions and avoid confusing symbols like .’.
  • check results of numerical computations without starting from scratch
  • fill important gaps in the mathematics curriculum of the UK schools

Our plan is to post at least once per month in the 4 broad categories:

  • dethroning a popular myth in the UK school mathematics
  • tips and common mistakes from our students’ experience
  • discussion of a general method or a powerful principle
  • educational story from our non-UK personal experience.

If you have suggestions for topics in our blog,  post a comment by clicking on “Leave a reply” in the upper left corner (name and e-mail needed) or e-mail blogger@master-maths.co.uk.

We hope to post weekly on Fridays, possibly fortnightly during vacations. Our next post will be on 5th July 2013: “How to get in top universities for maths degrees: 4 mistakes and 4 tips”.

For those who would like to test their maths skills right now, here is our first riddle:

  • Riddle 1: state the theorem illustrated by our logo in the upper left corner.
  • How to submit: to write your full rigorous answer, submit a comment.
  • Hint: the statement should be clear to anyone who hasn’t seen it before.
  • Warning: almost all first attempts are incomplete by our past experience.
  • 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 Learner 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.