Monthly Archives: December 2013

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.

length-round-arc 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.

4-geometric-objects 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.

Alan-Turing-photo 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.
  • How to submit: to write your full answer, click on “Leave a reply”.
  • 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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