Monthly Archives: September 2013

Next Big Thing in online education: 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.

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.

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.


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.

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 properly state mathematical theorems: 3 key parts

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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