Posted by: Alexandre Borovik | February 11, 2012

Cinderella and GeoGebra

[My ancient post (December 17, 2006) from a now defunct blog on Blogger: ]

In his comment on one of my previous posts, Euclid’s Elements, Interactive, Rick Booth brought my attention to GeoGebra, a free interactive geometry package. GeoGebra is nice, simple, intuitive to use, and could be a great help at school.

For some years, I was an occasional user of a commercial package, Cinderella. I decided to run a quick comparision of the two packages on a problem about the radical axis of two circles — I briefly mention the problem in my book. The problem is easy to solve with some help from Pythagoras:

Prove that the set of points P in the plane such that the tangents from P to two given non-intersecting circles are equal, is a straight line (it is called the radical axis of the two circles).

The problem is even easier when the two circles intersect – in that case, the radical axis is exactly the line through the two points of intersection. And in the case of tangent circles it is even easier — in that case the radical axis is the common tangent line.

It is interesting to compare the behaviour, in Cinderella and GeoGebra, of a simple interactive diagram: two interesecting circles of varying radii and the straight line determined by their points of intersection. In GeoGebra, when you vary the radii or move the centers of the circles and make the circles non-intersecting, the line through the points of intersection disappears — exactly as one should expect. In Cinderella, the line does not disappear, it moves following the movements of the circles, always separating them; when circles touch each other and start to intersect again, the line happens to be, again, the tangent line or the line through the points of intersection. And it is exactly the radical axis of the two circles!

I have not read documentation for GeoGebra, but the behaviour of this diagram suggests that, in GeoGebra (at least in the default mode), the underlying mathematical structure is the honest real Euclidean plane. In Cinderella, the underlying structure is the complex projective plane; what we see on the screen is just its tiny fragment, a real affine part. The radical axis of two non-intersecting circles is the real part of the complex line through two complex points of intersection; since the intersection points of two real circles are complex conjugate, the line is invariant under complex conjugation and therefore is real and shows up on the real Euclidean plane.

The Help Index of GeoGebra says about complex numbers:

GeoGebra does not support complex numbers directly, but you may use points to simulate operations with complex numbers.

However, even if one cannot do in GeoGebra the complex tricks of Cinderella, it is not a serious drawback — one should work hard to find a problem, of a secondary school level, where the difference in the nature of real and complex mathematical engines becomes visible. Such a problemis likely to involve circles or other conics, that is, quadratic equations — but remember, even the distance function is already involving a quadratic form, therefore complex numbers can pop up in rather unexpected places.

[I can add that another problem:

find the points of kiss of three mutually tangential cirles with centers at three given points

is linear, not quadratic, and can be solved by affine tools, say, with the help of a ruler with two parallel edges. ]

But still, when assessing software for mathematics teaching, it is useful to check the power, both computational and theoretical, of the mathematical engine. Unfortunately, most reviews of software for mathematics teaching never discuss this point.

To finish on a note more suitable for the festive season, I wish to mention that, in my book, radical axes appear in a solution of the following problem:

The Holes in the Cheese Problem. A big cubic piece of cheese has some spherical holes inside (like Swiss Emmental cheese, say). Prove that you can cut it into convex polytopes in such way that every polytope contains exactly one hole.

The previous discussion is the strongest hint for its solution.

And finally, a quote from Cheese.com:

“Emmental […] is considered to be one of the most difficult cheeses to be produced because of its complicated hole-forming fermentation process.”

Is it realy surprising that compex numbers naturally appear in the cutting the wheel of cheese?

Posted by: Alexandre Borovik | February 3, 2012

Antikythera Mechanism: A talk in Manchester

In April, the British Colloquium for Theoretical Computer Science will be held in Manchester.  The webpage.   Fees are quite reasonable (they essentially just cover the conference banquet, lunches and drinks provided).

We have a speaker, Mike Edmunds ,  involved with the fascinating investigation of the Antikythera Mechanism. Below is the abstract of his talk, The Antikythera Mechanism and the early history of mechanical computing:

Perhaps the most extraordinary surviving relic from the ancient Greek world is a device containing over thirty gear wheels dating from the late 2nd century B.C., and now known as the Antikythera Mechanism. This device is an order of magnitude more complicated than any surviving mechanism from the following millennium, and there is no known precursor. It is clear from its structure and inscriptions that its purpose was astronomical, including eclipse prediction. In this illustrated talk, I will outline the results – including an assessment of the accuracy of the device – from our international research team, which has been using the most modern imaging methods to probe the device and its inscriptions. Our results show the extraordinary sophistication of the Mechanism’s design. There are fundamental implications for the development of Greek astronomy, philosophy and technology. The subsequent history of mechanical computation will be briefly sketched, emphasising both triumphs and lost
opportunities……


Posted by: Alexandre Borovik | January 21, 2012

Exams and league tables

Nick Gibb MP, State Minister for Schools, published an article in the The Telegraph, outlining a reform of school league tables.

an article is followed by an interesting comment from JD:

If it is possible to “teach to a test” at the expense of a subject, then the fault is with the test.

They must simply be testing facts rather than any understanding.

I’ve discussed this with my engineers who inform me they had very  different finals for their Masters than I did.

The norm is multi choice and exams with 30 questions expecting fact recital.

From 30 years ago the emphasis was on very short questions expecting an essay for an answer. Classically this question would be to discuss methods to solve a problem that as yet hasn’t been solved (in this case using the engineering skills you’ve learnt).

I remember 30 years ago being asked one question with a 1 1/2 hour essay.

“Your director of engineering proposes a new product. A Digital Network Telephone with IP Packet based data. Discuss.”

This was years before TBL proposed the WWW – we were being asked to invent Skype in 1.5 hours.   This type of question isn’t possible to teach to – especially as the list of problems to be solved is endless.

In the same vein, I had to appraise a Navigation specialist whilst working offshore. He happened to be Malay (this was in Asia). He had a Masters Science degree. We asked him to solve a simple problem with the tools available offshore.

“The ships gyro has failed. How would you use the vessels onboard GPS systems and additional GPS to functionally replace the Gyro providing heading information”.

He couldn’t provide any answer (after two weeks). His excuse was there was no reference to this in any book. He said then the question wasn’t fair – how can you answer a question where you can’t find the answer in a book?

This is where education has gone wrong.

Posted by: Alexandre Borovik | January 16, 2012

Bertrand Russell on BBC

Archive on 4: Bertrand Russell – the First Media Academic?

Posted by: Alexandre Borovik | January 11, 2012

Mathematical Pasta

The NYT, Pasta Graduates From Alphabet Soup to Advanced Geometry by .

A rendering of pasta ioli, which George L. Legendre named after his daughter.

You got the idea. George L. Legendre’s book: Pasta by Design.

[With thanks to to muriel]

Posted by: Alexandre Borovik | January 11, 2012

Manual dexterity, II

This is a continuation of the previous post.

Among horrors proudly shown at the recent Computer Based Math Education Summit was a software package for primary school where pupils were supposed to enter numeric answers by moving, with a computer mouse, beads on a virtual number rack (two-string abacus) on the computer screen. Maria Montessori introduced number rack for enhancing pupils’ TACTILE perception of number!

Posted by: Alexandre Borovik | January 11, 2012

Manual dexterity

From an interview with Professor Heinz Wolff in the Engineering & Technology, an in-house magazine of IET.

After the brain, Wolff explains, the ‘most marvellous thing we have’ is the hand: an actuator that can thread a needle one minute or wield a sledgehammer the next without modification. ‘I firmly believe that the continual iteration of hand-eye-brain is how we became Homo sapiens.

‘We started to make tools, acquired manual skills and could imagine a tool that would be better. And then there was a very important point in our development, which was that we could imagine a tool that could make a tool, which could then make something. This is a very sophisticated way of thinking.’ His obvious implication is that this is the origin of engineering.

His hands flash across a QWERTY keyboard. ‘Apart from typing, we don’t use our hands. Girls don’t embroider; boys don’t play with Meccano. With these things you effectively develop an eye at the end of the finger, and you do this when you’re seven years old. And it’s really very clever. But it’s gone.’

Wolff has lectured on the ‘death of competence’ and he thinks it’s brought about by the abandonment of micromanipulation – doing something small and critical with the hand. ‘Our engineering students can’t make things. They might be able to design things on a computer, but they can’t make things. And I don’t believe that you can be an engineer properly, in terms of it circulating in your blood and your brain, without having a degree of skill in making things.’ He explains that this is why apprenticeships were so good, because ‘you actually made things while learning a bit of the theory’.

In neglecting to teach basic manual skills we are producing a generation that carries the seeds of its own impotence. Wolff believes that whilst all teachers agree children should be articulate and use language with precision and skill, ‘they don’t attach the same values to the use of their hands.’

Posted by: Alexandre Borovik | January 9, 2012

Alan Turing Centenary – Cryptography Competition

Alan Turing Centenary – Cryptography Competition

This starts officially on Monday 9th January and runs through until 16 April 2012.   There are already over 170 teams registered at the Cryptography website, with over 500 participating students:

http://www.maths.manchester.ac.uk/cryptography_competition_2012/

Please promote the competition wherever possible to ensure its success – we have some excellent prizes to be won sponsored by the flight search company, Skyscanner.

Posted by: Alexandre Borovik | January 7, 2012

Value-added scores, again

From NYT, Big Study Links Good Teachers to Lasting Gain, by Annie Lowrey:

Elementary- and middle-school teachers who help raise their students’ standardized-test scores seem to have a wide-ranging, lasting positive effect on those students’ lives beyond academics, including lower teenage-pregnancy rates and greater college matriculation and adult earnings, according to a new studythat tracked 2.5 million students over 20 years.

And some sobering remarks:

Still, translating value-added scores into policy is fraught with problems. Judging teachers by their students’ test scores might encourage cheating, teaching to the test or lobbying to have certain students in class, for instance.

“We are performing these studies in settings where nobody cares about their ranking — it does not change their pay or job security,” said Jesse Rothstein, an economist at the University of California, Berkeley, whose work criticizing other value-added assessments unions frequently cite. “But if you start to change that, there is going to be a range of responses.”

I personally would propose an alternative way to assess school teachers: by their students’ performance at the next stage of education. For example, the best criterion to judge a GCSE level mathematics teacher is to look at numbers and academic performance of those his/her students who choose Mathematics/Further Mathematics as an A Level subject.

[With thanks to muriel]

Posted by: Alexandre Borovik | January 6, 2012

Finland’s School Success

What Americans Keep Ignoring About Finland’s School Success, an article by Anu Partanen in The Atlantic.

A quote:

For starters, Finland has no standardized tests. The only exception is what’s called the National Matriculation Exam, which everyone takes at the end of a voluntary upper-secondary school, roughly the equivalent of American high school.

Instead, the public school system’s teachers are trained to assess children in classrooms using independent tests they create themselves. All children receive a report card at the end of each semester, but these reports are based on individualized grading by each teacher. Periodically, the Ministry of Education tracks national progress by testing a few sample groups across a range of different schools.

As for accountability of teachers and administrators, Sahlberg shrugs. “There’s no word for accountability in Finnish,” he later told an audience at the Teachers College of Columbia University. “Accountability is something that is left when responsibility has been subtracted.”

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