I have every good reason to recommend watching “The Man Who Knew Infinity”, an account of the life of the Indian mathematician Srinivasa Ramanujan.

Ramanujan was an amazing figure who had practically no formal training in Mathematics, yet made extraordinary contributions to various fields of the subject. The film centres on his years at Trinity College, Cambridge University, where he worked closely with the English mathematicians G. H. Hardy and J. E. Littlewood.

The film can easily be followed by non-mathematicians. I particularly liked the scene where Ramanujan tries to explain to his wife (and to the general audience) that his love for Mathematics comes from the tendency of mathematical patterns to appear in ways that cause surprise. Hardy also contributes to this by explaining to his butler what Ramanujan was trying to do when tackling the problem of partitions.

The film presents us with some lovely poetic images of India and I cannot forget that scene where Ramanujan’s wife looks on, as her husband sails away in the boat that will take him to a ship bound for England.

Jeremy Irons gives a brilliant performance as the cantankerous G. H. Hardy and at the end of the film quotes from Hardy’s famous and haunting book A Mathematician’s Apology. Dev Patel is also very convincing as the youthful and enthusiastic Ramanujan. The culture shock that he experienced at Cambridge is illustrated well and one can only feel sorry for him, as well for the fact that his life ended so soon.

The only departure from historical fact that I managed to pinpoint concerns the exchange between Ramanujan and Hardy concerning the number 1729. The film shows this as taking place when Hardy bids farewell to Ramanujan, as the latter sets off on his return journey to India. In fact, the exchange took place when Ramanujan was in hospital.

In closing, I must again say ‘bravo’ to the filming world for yet another good film about Mathematics and mathematicians. It has already given us ‘Agora’, ‘A Beautiful Mind’ and ‘The Imitation Game’. I should also include ‘The Theory of Everything’, as Stephen Hawking has used Mathematics so much in his explorations as a cosmologist.

posthumous prize = awarded to one after one’s departure from this world
post-hummus prize = awarded after eating that particular delight

Recently, the following question was set in a GCSE maths exam:

Liz buys packets of coloured buttons.

There are 8 red buttons in each packet of red buttons.
There are 6 silver buttons in each packet of silver buttons.
There are 5 gold buttons in each packet of gold buttons.

Liz buys equal numbers of red buttons, silver buttons and gold buttons.

How many packets of each colour of buttons did Liz buy?

A certain student answered by calculating 8 × 6 × 5 = 240, concluding that this was the number of beads of each colour and then deducing that Liz bought 30 red, 40 silver and 48 gold buttons.

The student lost marks, apparently because the instructions given to the examiners stated that the answer, that had to be arrived at for the number of buttons, of each colour was 120.

Yet it is the question that is at fault, because although 120 is the least common multiple of 8, 6 and 5, the wording does not include any information pointing to the fact that it is this, which is being sought. In fact any common multiple will do, hence the easy choice of 240.

This is quite unfair on the student, but the damage most probably cannot be undone.

There are better ways of phrasing the question and I have thought of one, which in my view also serves to instruct and teach, besides stating the question for the purposes of the exam:

Liz buys packets of coloured buttons.

There are 8 red buttons in each packet of red buttons.
There are 6 silver buttons in each packet of silver buttons.
There are 5 gold buttons in each packet of gold buttons.

Liz wishes to buy the same number of buttons of each of these three colours and, in such a way, so as to obtain the least possible number of buttons.

Calculate how many packets of each colour of buttons Liz should buy, the number of buttons of each colour thus bought and the total number of buttons.

I believe this way of phrasing the question has some benefits:

  1. It illustrates a more natural way in which the problem could occur in real life (Why miss an opportunity to bring Maths a little closer to our daily lives for the sake of students?)
  2. The last sentence helps the student make a distinction between the number of packets and the number of buttons. This is important, because one quite easily could mistake packets for buttons.

And here is a solution to the problem:

Whatever the number of packets of red buttons Liz chooses, she will always end up with a multiple of 8. Similarly, she will end up with a multiple of 6 for the silver buttons and a multiple of 5 for the gold buttons.

Therefore, if Liz wants the same number of buttons of each colour, this common number must be a common multiple of 8, 6 and 5. Since she wants this number to be the least possible, it follows that it must the least common multiple (LCM) of these numbers.

To calculate it, let’s decompose these numbers into prime factors:

8 = 2³

6 = 2 × 3

5 is prime

Hence the LCM of these three will be 2³ × 3 × 5 = 24 × 5 = 120.

[Digression: If one is not happy with this calculation, then one might consider that LCM(8, 6, 5) = LCM(LCM(8, 6), 5).

For the LCM of 8 and 6, we list the multiples of these two:

for 8: 8, 16, 24, …

for 6: 6, 12, 18, 24, …

from whence it follows that LCM(8, 6) = 24. The multiples of 24 and 5, now, are:

24: 24, 48, 72, 96, 120, …

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125…

and the result follows.]

So, Liz must buy 120 buttons of each colour. To achieve this, she will have to buy:

120 ÷ 8 = 15 packets of red buttons

120 ÷ 6 = 20 packets of silver buttons

120 ÷ 5 = 24 packets of gold buttons

The total number of buttons is 3 × 120 = 360.




Today, in #Eidomeni, Northern Greece, on the border with FYROM, there are around 14000 refugees; 7000 of them are children and 70% of these children are ill.
The camp, which has been set up there can only hold 4000 people, which means that 10000 are most probably living “outdoors”. To keep warm, they burn everything that they can, including plastic, which means that the air is polluted.
The local community chief said that the locals would like to see the 10000 surplus refugees leave, because they are currently occupying their farmland, which is thus rendered unusable.
The total number of immigrants in Greece today is around 36400. Out of these, 90% are refugees and 10% are not.

I am pleased with the latest Star Wars instalment, “The Force Awakens”. I found it to be good, clean Star Wars fun… I was impressed by the fact that roughly the first half hour of the film is laden with scenes of pursuit!

[Spoiler Alert for what follows below]

Homage to the earlier films

There are a few themes in the film that seem to carry over from the earlier six films:

  1. Again, the fate of a galaxy seems to hinge on the relations between the members of a family. This time, we have a turnaround, for it is now the son – as opposed to the father – who has been led astray.
  2. Again, a droid is made to carry a vital set of information.
  3. Again, the hero leads a difficult life on a remote planet and is catapulted into action by a chance meeting with this droid. The hero is initially oblivious to the powers that they have.

I don’t mind this repetition, in fact it is probably the very basis of every good fairy tale. Even so, I think the film made a respectable effort not to repeat too much of its predecessors.


Harry Potter and “The Force Awakens”

I noticed that two concepts seem to have been “borrowed” from the Harry Potter realm:

  1. The fact that a lightsabre seems to “seek” its rightful owner, which was never touched upon in the previous six films and is akin to the situation where a magical wand chooses its owner.
  2. The ability to use the Force to read one’s mind, in much the same way as Voldemort peers into Harry’s mind.


The Big Question: Rey and Luke

Many have been speculating on the relationship between Luke Skywalker and Rey. From the trailers that were released prior to the release of the film, as well as the issue with Luke’s lightsabre choosing Rey instead of Kylo Ren in the snow, I would guess that Rey is indeed Luke’s daughter. I imagine that she was abandoned on Jakku when Luke went into hiding, so that she would not be discovered by the First Order.

Rey also appears to give a hint towards that being the truth: When Maz Kanata talks to her about “the family you long for”, Rey whispers “Luke”.

It is likely that Rey has visited the island where Luke is hiding, because Kylo Ren saw an ocean when he looked into Rey’s mind.

We shall have to wait and see how things turn out. What also remains to be seen is how Maz got hold of the lightsabre in the first place.

While we’re on the subject, the “helicopter shot” of Luke and Rey on Skellig Michael, at the very end of the film, was fabulous.


More Interesting Points in the Film

  1. It is likely that Rey has been inside the Millenium Falcon before the events of the film. She knew how to pilot it (more or less), as well as where the position of the gunner was. However, she expressed surprise when Han Solo referred to the ship by its name, so it seems that she didn’t know the identity of the ship.
  2. It is a pity that Han Solo died in the film, but this is understandable, since the new films will have to make way for their new heroes. I expect that Leia, Chewbacca and Luke may die in the next instalments as well. I should mention that I read somewhere that Harrison Ford had pushed George Lucas to have Han Solo killed in “Return of the Jedi”, but Lucas refused. Well, Ford got his way this time…
  3. It was interesting to see the Millenium Falcon receive such a pounding on Jakku and Starkiller Base… and a pity to see Chewie piloting the Falcon alone after the death of Han Solo (this for the first time ever in a “Star Wars” film).
  4. A long time ago (in this galaxy!), I read that George Lucas had initially planned for the hero to be a girl and then changed over to Luke. Well, there we are then, his original intentions have been realised.
  5. Luke’s original name in Lucas’s script was Starkiller not Skywalker, so the name of the First Order’s base must be a nod to Lucas’s original ideas.
  6. Finn is the very first Stormtrooper (not Clonetrooper) ever to be seen without a helmet.
  7. This film is the first to show TIE fighters inside a hangar and alongside their pilots. In the other films, we saw them only in space.
  8. Did Supreme Leader Snoke appear as a hologram? Because, in one scene, he seemed to disappear after the end of a conversation.
  9. I liked the parallel between Rey and Finn: They both grew up far away from their families.

Enjoy the awakening of Star Wars and sit tight until next year for Episode VIII!

Until then, may the Force be with You!

Σήμερα πήγα στην συνέντευξη τύπου που έδωσε η Συντονιστική Επιτροπή Συλλόγων Γονέων Ιδιωτικών Σχολείων με θέμα την ακύρωση της επιβολής ‪ΦΠΑ‬ στην ‪ιδιωτική‬ ‪παιδεία‬.

Κατά την διάρκειά της αναπτύχθηκαν πολλές σωστές απόψεις. Μνημονεύω τα κύρια σημεία:

1. Το σύνολο των γονέων που έχουν παιδιά σε ιδιωτικά σχολεία δεν αποτελείται εξ’ ολοκλήρου από “έχοντες και κατέχοντες”, ώστε να δικαιολογείται η επιβολή ΦΠΑ στα δίδακτρα. Οι “έχοντες” αποτελούν μόνο το 15% αν δεν έχουν ήδη πάει τα παιδιά τους σε σχολεία του εξωτερικού. Την πλειοψηφία αποτελούν μέσες οικογένειες των οποίων ο προυπολογισμός θα επιβαρυνθεί πολύ με την επιβολή του ΦΠΑ.

2. Οι γονείς με παιδιά σε ιδιωτικά σχολεία ήδη πληρώνουν φόρους που στηρίζουν την δημόσια παιδεία, ενώ ουσιαστικά δεν κάνουν χρήση αυτής και έτσι δεν επιβαρύνουν το κράτος. Επισημάνθηκε ότι η δαπάνη των διδάκτρων αποτελεί τεκμήριο στην φορολογική τους δήλωση.

3. Όσοι γονείς δεν αντέξουν τις αυξήσεις στα δίδακτρα λόγω του ΦΠΑ θα υποχρεωθούν να βάλουν τα παιδιά τους σε δημόσια σχολεία στην μέση της χρονιάς κι αυτό θα έχει αρνητικές επιπτώσεις πάνω στην ψυχολογία των παιδιών. Μεταξύ άλλων, θα διακόψει την προσπάθεια κοινωνικής ένταξης που ήδη έχει ξεκινήσει το κάθε παιδί. Επίσης, θα προκαλέσει απολύσεις σε βοηθητικό προσωπικό των σχολείων (οδηγοί, καθαρίστριες κ.α.), οδηγώντας πολλούς στην ανεργία και επιβαρύνοντας τον ΟΑΕΔ. Τα παιδιά που θα πάνε σε δημόσια σχολεία θα επιβαρύνουν το κράτος, που θα πρέπει να δαπανήσει για αυτά.

4. Θεωρείται αυτονόητο από την Επιτροπή ότι ο Πρωθυπουργός θα υλοποιήσει την προεκλογική του υπόσχεση για την ακύρωση της επιβολής του ΦΠΑ. Αυτό ελπίζει η ίδια (και εγώ λέω “από το στόμα σας και στου Θεού τ’ αφτί”)

5. Έχει ζητηθεί συνάντηση με τον Υπουργό Παιδείας κο Φίλη για το θέμα, αλλά ο ίδιος δεν δέχεται κανέναν μέχρις ότου γίνουν οι προγραμματικές δηλώσεις της Κυβέρνησης.

Ύστερα από αυτήν την σύνοψη, επιτρέψτε μου να προσθέσω την δική μου άποψη:

Κε Πρωθυπουργέ, ο δρόμος που πρέπει να ακολουθήσετε είναι προφανής: Διασώστε την αξιοπιστία σας, ακυρώνοντας αυτήν την βλαβερή επιβολή.

ΥΓ: Για όποιον ενδιαφέρεται να επικοινωνήσει με τον σύλλογο, υπάρχει η ιστοσελίδα www.sylgonis.gr

Μία ζεστή βραδιά μας επεφύλαξε η κα Ελευθερία Βαρελά – Νεραντζοπούλου, δασκάλα μου στην 1η και 2α δημοτικού, χτες Δευτέρα 4/5, στο βιβλιοπωλείο “Ιανός”.

Σε μία όμορφη εκδήλωση μας παρουσίασε το πρόσφατο βιβλίο της “Θυμάσαι; Οι αναμνήσεις μιας δασκάλας”. Μαζί με την ηθοποιό Κάρμεν Ρουγγέρη μας διασκέδασε με τα καμώματα, τους προβληματισμούς και τις εκφράσεις μιας μεγάλης πλειάδας παιδιών που συνθέτουν, σαν τα κομμάτια ενός παζλ ή κολλάζ, την μακρόχρονη διδακτική της πορεία.

Πρέπει να πω πως ήταν μεγάλη και αναπάντεχη τιμή για μένα που η συγγραφέας ξεκίνησε το βιβλίο με τρεις σελίδες αφιερωμένες σε μένα, τον μικρό Άντυ, όπως ήμουν γνωστός τότε σε δασκάλους και συμμαθητές…

Το βιβλίο έχει εκδοθεί από τις εκδόσεις “Διάπλαση”: http://www.diaplasibooks.gr/product-297.html

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