How and when school children learn the times table

This post was motivated by a long public discussion of learning the times table at UK schools.

Dropping maths at the age of 16

It seems that a public acceptance of being bad in maths may not be considered as a great shame, even by a finance minister, read BBC magazine: UK Chancellor refused to answer a “times table” question. How could that happen?

One retired maths teacher has told us that many students do not continue studying maths and English at the age of 16-18. Here is an update from BBC education: a long-term possibility might be to require all would-be teachers to study maths to 18.

This long term plan can be compared with the long existing tradition in other countries where maths and native language/literature are compulsory for all students up to the university level.

Why are UK students really lucky?

Here is the most popular comment on BBC: “I’m studying a Masters in Physics at University and STILL don’t know my times tables by heart, I got an A* at GCSE Maths and an A at A-Level.”

Does it imply that UK students may not know the times table? It is probably better not to reflect on conclusions that overseas students can make about UK qualifications and universities.

Here is our reply: you are really lucky to study in the UK, because without the times table you wouldn’t progress to a secondary school in a different country.

In many other educational systems, if children start school at the age of 6-7, then they finish learning the times table by the end of their 1st class at the age of 7-8.

Why is learning the times table so hard?

As with all other obstacles, learning mathematics becomes unnecessarily hard when pre-requisite concepts are missed. The necessary earlier step is the simpler addition table that has the sum i+j in the intersection of the i-th row and j-th column.

 + 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 14 15 16 17 18

Notice how the table above is beautifully symmetric like many great results in mathematics. When students learn the times table not by rote, but by actually trying to compute products in their head, they naturally start adding smaller numbers.

For instance, to find $$7\times 8$$, one can sequentially compute the following partial sums: $$8+8=16$$, $$\;16+8=24$$, $$\; 24+8=32$$, $$\; 32+8=40$$, $$\; 40+8=48$$, $$\; 48+8=56$$. These sums require the addition table above, which should be studied before the times table.

So we get $$7\times 8=56$$ after completing 6 easy additions. If we are already good at multiplying only by 2,  faster way is to do 3 multiplications: $$7\times 2=14$$, $$\; 14\times 2=28$$, $$\; 28\times 2=56$$.

Can the times table be learned only by rote?

Let us compare the times table with another basic skill of walking. Children usually start learning to walk earlier than learning the times table.

Toddlers fall (fail) many times before they become confident walkers. After a first fall a baby looks at their parent to understand how to react: if the parent looks afraid then the baby starts crying, if the parent smiles then the baby also smiles.

So almost all children successfully learn to walk, not by rote, but by actually trying to walk, despite numerous falls or even small injuries. If a kid can not walk by the age of 3, it is usually considered as a physical disability.

A criterion of success in learning

In our opinion, the same powerful principle works for learning any other basic skill. So we wish our readers to follow the excellent example of smart babies, namely more practice and consider every failure (or a fall) as one more step in a right direction.

There is even the internal criterion of success without any comparison with your peers. If people can walk, they walk easily and without making any conscious efforts. Similarly, multiplying single-digit numbers should be as effortless and enjoyable as walking or breathing.

• Riddle 17: can the square $$n^2$$ of an integer have the last digit 8?
• How to submit: to write your full answer, submit a comment.
• Hint: the required knowledge is (a small bit of) the times table.
• Warning: give an example or justify that there is no such number.
• Prize: free 1-year access to one of our interactive web tutorials.
• Restriction: only the first correct public answer will be rewarded.

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Cultural differences: failure vs life-changing experience

Happy New Persian Year Nowruz if you are celebrating on March 21!

In our previous post on common mistakes with square roots we have dethroned the popular myth that $$\sqrt{x^2}=x$$. Below we shall tell our readers a personal story how we gained a life-changing experience due to the basic fact that $$\sqrt{x^2}=|x|$$. Students have told us that the absolute value $$|x|$$ is taught in the UK only at A levels (age 17-18). Moreover, the last year STEP examiners’ report again highlighted this common mistake about $$\sqrt{x^2}$$. So we have shown the graph $$y=|x|$$ in the picture above just in case.

Entrance exams in Physical-Mathematical Schools

We started to learn maths at overseas schools that specialise in mathematics and physics. All such schools are state-funded, but have rigorous entrance exams, usually at the age of 12 or 13. Our traditional riddle below contains Question 2 (of 8) from a recent entrance exam.

So we were in a class of 30 best maths students selected in a 1-million city. Everyone was a maths champion in their previous school. However our new Physical-Mathematical School was rather different. Grades were not given for past successes or for social connections, but only for hard independent work.

Answering in front of a class

Solving unexpected problems at a blackboard in front of a class was a routine exercise every lesson unless we had a written paper for independent solving. Once the teacher called one of the prettiest girls, who might have been named as a class queen in the UK. At that time she actually was in the top 20% of the class due to her background knowledge from previous years.

While the girl was trying to solve a new problem at the board, our teacher realised that the girl confuses some basic things. So she directly asked: “What is the square root of $$x^2$$?” The girl replied: “x”. The teacher: “That’s a fail, sit down.”

There was complete silence, nobody laughed. Indeed, despite we were the best in our previous schools, I’m sure that everyone from our class had been in a similar sutation before, probably with more advanced stuff than $$\sqrt{x^2}$$, however we understood all feelings anyway. The key lesson in this story is not about $$\sqrt{x^2}$$ at all, but about the exemplary reaction to such a failure.

The girl hold the nerves and quietly sat down and was actually ok after the lesson. The whole class was really impressed by her level of self-control at the age of 13. I’m not sure if it was a life-long lesson for the girl, but definitely for me since I can vividly recall all the details more than 20 years after this lesson. Later when I was in many tough situations I remembered how to properly withstand a punch with dignity and never give up!

There is no success without failures

Without our failures at school, we wouldn’t become professional mathematicians, but this is another story for our future post. The key arguments are outlined in the article The science of success by NewScientist. You may also enjoy the recent success story of the new billionaires behind WhatsApp from BBC. Here is a couple of the very inspiring quotes: “Got denied by Twitter HQ. That’s ok. Would have been a long commute.” “Facebook turned me down. It was a great opportunity to connect with some fantastic people. Looking forward to life’s next adventure.”

It is impossible to imagine in the UK anything similar to the school story above. Students are usually not called to solve unexpected problems in front of a class, even at university. Many vice-chancellors are proud of a 2% dropout rate at their universities. At the opposite extreme, we can mention Bear Grylls, who was one of 4 (among 180) candidates selected for SAS.

Weeping because of Pythagoras’ theorem

When we were running summer schools for the gifted and talented in the UK, we knew that students should not be asked, at least if they prefer to hide. Once we were discussing Pythagoras’ theorem with year 9-11 students (of age 14-16). Most students knew at least the first half of full Pythagoras’ theorem even if without a proof. However, one girl who was hiding quite well unfortunately decided that Pythagoras’ theorem is beyond her ability and started to cry.

This accident has taught us that we should teach Pythagoras’ theorem in  a much more careful way. That is why we have designed our free tutorial on Pythagoras’ theorem including its full statement, proofs in both directions and applications to diaphontine equations. Actually, our riddle on a proof of converse Pythagoras’ theorem is still open for more than 6 months. Hence those students who have enough motivation to self-register and master our free tutorial, can still solve this Pythagoras’ riddle and gain access to other tutorials on advanced topics.

Your personal choice: crying or learning

You could make your own conclusions from the cultural differences described above. We shall highlight only our own personal choice. We are human and also make mistakes, which is the human nature. Even robots fail and actually more frequently than humans. However, every failure makes us only stronger, because we learn not to repeat the same mistake again. And the harder we fail, the stronger we become. So we call all our failures the important life-changing experience. Here is the powerful principle: the harder the training, the easier the exam!

Our riddle is from an entrance exam to a Physical-Mathematical School at the age of 12-13.

• Riddle 13: factorise the polynomial $$6m^4-3m^3-4m^2+1$$.
• How to submit: to write your full answer, submit a comment.
• Hint: any factor of a polynomial gives a root and vice versa.
• Warning: justify that you found all factors with real coefficients.
• Prize: free 1-year access to one of our interactive web tutorials.
• Restriction: only the first correct public answer will be rewarded.
• Update: Chen has excellently solved the riddle at first attempt.

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The unlucky deduction: mathematics vs mathology

We wish a Merry Christmas and a Happy New Year 2014 to all mathematics enthusiasts!

The unlucky deduction in teacher training

This post is motivated by few news stories. The first one is a personal story from our friend who is completing a teaching qualification and had some practice in a UK private school.

On one day the future maths teacher was discussing many examples with a class how to find the length of a round arc using the angle at the centre. For instance, if r is the radius of the circle, the full angle 2π corresponds to the full circumcircle of length 2πr, where . The angle π gives only the half-circumcircle of length πr and so on.

We should mention that our friend has completed 5 years of proper undergraduate maths studies in another country, where students do not expect a lot of help from their lecturers and would consider such basic examples as a strange spoon feeding at the age of 13-14.

On the next day the hopeful trainee decided to revise and introduced the formula βr for the length of a round arc with a radius r and an angle β (in radians). Then the pupils were asked to refresh the yesterday computations by substituting simple angles β to get the required length of the round arc. At this moment the observer who was a qualified teacher immediately stopped the lesson and explained that in this country the maths teachers are not allowed to use general statements for deducing more specific results.

Deductive reasoning in mathematics.

Making a logical conclusion in a specific case from a general theorem is called deductive reasoning, or simply deduction (not subtraction). Mathematical induction is (in a sense) the opposite approach when a general claim is proved via easier base and inductive step.

If the deduction is forbidden, it seems very logical that most British students unfortunately struggle with a full statement of Pythagoras’ theorem, let alone its proof in both directions. That is why we designed our free web tutorial on Pythagoras’ theorem and applications.

The Britain is world-wide known for its famous Sherlock Holmes whose crime stories are actually based on deductive reasoning. Reading a few novels by Arthur Conan Doyle may help the authors of the UK school curriculum improve the career chances of UK students.

Deduction was accepted in Ancient Greece.

We are not professional novelists, but could tell a short mathematical story how deductive reasoning was formalised in Pythagoras’ school more than two thousands years ago in Ancient Greece. The word “geometry” is combined from the Greek words “geo” (earth) and “metron” (measurement). If you start to measure a square piece of land (as ancient Greeks certainly tried), the first obstacle is to get the ideal right-angle 0.5π (90 degrees).

The students who are familiar with full Pythagoras’ theorem could quickly give the practical recipe: measure the lengths of sides a,b,c of a triangle, if a2+b2=c2 then the angle opposite to side c is right (90 degrees). So converse Pythagoras’ theorem (not the usual direct theorem) was practical, because measuring lengths by a standard stick was easier than angles.

Using the basic deductive reasoning, we can conclude from direct Pythagoras’ theorem that the right-angled triangle with two smaller sides 1 has the longer side $$\sqrt{2}$$ (the square root of 2). This expression looks obvious for us now, but ancient Greeks operated only with rational numbers. Indeed, using fractions of a stick or a rope, they could get any rational number, but they couldn’t really measure $$\sqrt{2}$$. This “mysterious” number led to a philosophical crisis.

Does Ofsted hide the deductive reasoning from pupils to avoid such a crisis in British schools for health and safety reasons? The government will not worry, because most UK students take a calculator and use an approximate value, say 1.414213562373. However, in real life exact computations are much more valuable. Victims wouldn’t be happy after a fatal accident caused by the fact that 1.4142135623732<2, for instance in the space industry.

Mathematics vs mathology: one example.

Ancient Greece should be famous not only for widely known Greek mythology, but also for initiating rigorous logic, for proving first formal theorems and for making justified conclusions by deductive reasoning. We haven’t made a typo in the heading , but propose the term mathology for all improper numerical practices contrary to rigorous mathematics. So the comparison of mathematics vs mathology is very similar to astronomy (real science) vs astrology (rubbish pseudo-science). Mathology ignores the deductive reasoning.

A more specific example is the following question that is popular even in entrance tests at UK grammar schools. A few (usually four) objects are drawn and students are asked to select the odd one. This “problem” is mathological, because without extra restrictions any of four objects can be considered as the odd one by a properly justified mathematical argument.

These mathological “problems” are easily marked in a multiple-choice form and may not be suitable for bright individuals  who can “think outside the box”. As a result, parents pay special tutors who show their children how to guess standard patterns and pass mathological tests. A proper mathematical problem would be to find four different reasons when each of the given objects can be considered as the odd one, see our riddle below.

Congratulations on the PISA results!

According to the facts, the UK gained good international ranks, read BBC Education: PISA test (Programme for International Student Assessment): rank 26 in maths, rank 23 in reading, rank 21 in science. It remains to hope that the next results can be only better in 2016.

We wish a long life to all maths heros!

Finally, we highlight the real British Hero, read BBC Technology: royal pardon for codebreaker Alan Turing. Alan Turing is now considered as the father of modern computers, but was convicted for homosexuality in 1952, punished by being chemically castrated, lost his security clearence for code-breaking work and finally committed suicide in 1954.

In our opinion, by breaking the German code of the Enigma machine, Alan Turing contributed much more to the victory over nazis in the World War II than most generals and politicians. In fact, Churchill was the UK prime minister in 1940-1945 and then again in 1951-1955 and could had known about Alan Turing’s impact on the world history.

Here is the highest rated comment on the BBC news article (222 votes): “Somehow this seems the wrong way round – her majesty’s government should be asking for a pardon over Turing’s treatment, not granting one sixty years too late.” It is unfortunate, but not surprising, that the computer industry flourished after Alan Turing in another more tolerant valley, not around the Thames river. So we wish all contemporary mathematics enthusiasts to live long enough
and get rewarded for their contribution to the world knowledge (and peace).

• Riddle 10: find 4 different reasons when each of 4 geometric objects
(cycle, disk, square, sphere) in the picture above can be the odd one.
• Hint: each of four reasons can be a geometric property
that distinguishes one object from the remaining three.
• Warning: as usual in mathematics, but not in mathology,
there are many different correct arguments (four reasons).
• Prize: free 1-year access to one of our interactive web tutorials.
• Restriction: only the first correct public answer will be rewarded.

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What is important for success: 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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Introduction to our Master Maths blog: 4 reasons to start

We 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.

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.