Thursday, 14 September 2017

Badly-Written Maths Questions [BODMAS]

I have ranted in the past (albeit briefly) about badly written maths questions. These are the kind of question that do the rounds on Facebook, where - due to the deliberately ambiguous way that the question is written - there are at least two different answers.

The idea of these questions isn't to test people's maths skills.  It's designed to 'go viral' by generating conflict and disagreement between the know-it-alls, the qualified mathematicians and those who can't recall or don't know how to handle maths questions when there isn't enough information to easily proceed.

You know the kind of thing:

What is 3 + 4 * 5 + 6 - 7?
Only 1 out of 10 will get this right!

Firstly:  this is NOT proper Maths.  It just isn't.  Don't worry if it's confusing - it's deliberately intended to be.

Secondly:  if you don't get it 'right', then you'll probably continue feeling that maths is irrelevant, complicated, meaningless and inaccessible.  Because that's probably what you thought before, and the long list of comments that say it's 22  or 34, all equally convinced that they're right and the other person is wrong.  There'll be a few comments about showing how they've worked it out, and then a few people will say BODMAS.


BODMAS is the agreed way in which we carry out calculations like my example.  Mathematicians don't like uncertainty or ambiguity, and will go to great lengths to make their meaning perfectly clear and precise.  All scientists are the same - they show great precision in language, whether that's words or numbers.

BODMAS states that a calculation should be carried out in a particular order:

Brackets - any terms in brackets (or parentheses) should be calculated first.
Orders - any numbers which are raised to powers (squared, cubed, square root) are done next, after any calculations in brackets.  (Previously called Operators)
Division - divisions are the next priority.  Any two numbers or terms which are next to each other have to be divided, after any brackets and operators, but before anything else.

Multiplications - after you've done all the divisions, you then do all the multiplications.
Additions - any terms which are to be added together are done after the multiplications.
Subractions - finally, any remaining amounts are to be subtracted.

So, to take my example:

3 + 4 * 5 + 6 - 7  =  ?

There are no brackets or orders (powers) in my calculation, so the first calculation I will do is the Multiplication.  4 * 5 = 20.

So now, my calculation looks like this:

3 + 20 + 6 - 7 = ?

There's no dividing in my expression, so I can move on to the additions:  3 + 20 + 6 = 29

Which leaves me with:
29 - 7 = 22

And the answer is therefore 22.

While it may be possible to read the question differently, this will give a mathematically inaccurate [wrong] answer.  It may seem natural to read the question from left to right, but this will give a different and wrong answer:

3 + 4 (=7)
*5 (=35)
+6 (=41)
-7 = 34

Wrong answer = 34.

If you think that it's unfair or unrealistic to have to follow such precision, let me present some examples from written English, that show how important it is to state things clearly and in the right order:

I'm glad I'm a man, and so is Lola
Is Lola a man?  Are you reading from left to right, or did you go back to the middle?
He fed her cat food.
Was he looking after her cat?  Or was he making a culinary error?
John saw the man on the mountain with a telescope.
Who has the telescope?

Or how about this one, which has recently started going around Facebook, and is (almost certainly deliberately) full of mathematical and grammatical problems.

1 rabbit saw 6 elephants while going to the river.
Every elephant saw 2 monkeys going towards the river.
Every monkey holds 1 parrot in their hands.
How many Animals are going towards the river ???
Does "to the river" count as "towards the river"?
"Every elephant saw 2 monkeys" - is that each elephant saw 2 monkeys, or they all saw the same 2 monkeys?
"How many animals?" - this depends on if you include birds in your definition of animals (some do, some don't).
This is the epitome of a trick question, and this kind of uncertainty is completely unacceptable in maths - but that's what drives the apparently viral threads on Facebook.  People will argue vehemently about one answer or the other - confusing everybody else and leading to the frustration that we see (it's much easier to explain things in a five minute conversation than it is with five paragraphs of comment text on social media).

Maths has enough of a bad reputation for being confusing, inaccessible and frustrating; it doesn't need people asking "What's 5 + 6 *7 -8? Only 1 in 5 know the real answer!" to make it any worse.

(The answer is 39)
(The answer to the Albert Einstein question (which is particularly devious)  is -13
3 - (6*3) + 2 = 3 (- 18 + 2) = 3 - 16 = -13

Tuesday, 12 September 2017

Doctor Who: Warriors of the Deep

Doctor Who: Warriors of the Deep, on DVD

I've recently acquired a number of classic Doctor Who episodes on DVD from my local Salvation Army charity shop, and have decided to watch and review them here - under the category of 'miscellaneous opinions'.

The first of these DVDs is "Warriors of the Deep", featuring Peter Davison as the Doctor, in a series that originally aired in January 1984.  To summarise the plot: the Doctor and his two companions, Tegan and Vislor, materialise on Sea Base 4, in the middle of a tense situation between the humans and the 'warriors of the deep'.  Sea Base 4 is deep under the sea, isolated from outside contact, and as we join the story we discover that they're  monitoring their enemies' movements.  The enemy has stealthily moved a probe to within observation range of Sea Base 4, but they've still be seen. Sea Base 4 wait, and then release a probe to investigate further.  This quiet move and counter-move is a recurring theme throughout the story, as the tension is slowly ratcheted up.

While the plot between Sea Base 4 and its alien opposition unfolds, we discover that the base's medical officer (Doctor Solar) and its controller have deliberately maneouvred an inexperienced officer into the role of the base's weapons officer.  As the pressure and stress of the outside situation increase, the rookie officer starts to crumble, and the controller and medical officer are able to manipulate the base commander into giving them a duplicate controller disk.  The purpose of this isn't immediately clear, but something is clearly awry.  Conversely, the underwater aliens have a clear purpose - to resurrect or awaken a large army of dormant warriors who've been stored in an underwater cavern for many years.

The story moves along steadily but not quickly - and with a constant sense of tension in the atmosphere.  The sets and costumes are understandably dated (filmed in 1984) but once you look past this, it's an engaging story.  The Sea Base has been conceived as part of a sterile,white and metallic future, with bright, clean lines throughout (similar to the Tardis of its day).  The main questions that the story explores are: are the aliens more dangerous than the duplicitous humans; and who are the aliens anyway?  

While moving along, the story gives a remarkably high proportion of time to the supporting cast - the Doctor and his companions barely get around 15% of the screen time in the first episode (by my estimation), and the second episode is the same.  I guess it's a reflection of the mini-series approach of the classic series, where  more time could be dedicated to building up the scenario and the various characters with their own stories and plans.  The story takes its time, but doesn't seem to dawdle, as each of the characters follows their own arc.

There are the overtones of the Cold War - mutual trust between the Doctor and the Sea Base officers, mutual distrust between the humans and the underwater aliens - as was common in 1980s' TV series.  It's genuinely difficult to decide whether to take sides with the aliens or the humans; neither side acts with complete transparency.  The aliens move first, and despite the Doctor's warning the humans fire first - to no avail.  The Silurians claim to be fighting a 'defensive war' (the Doctor claims there is no such thing) and are manipulating the humans into fighting a destructive war amongst themselves. It's a novel idea, and it certainly makes sense given the humans' duplicitous behaviour throughout this story.  The tension continues to increase as the Silurians begin their plot to initiate planetary war among the humans, and the Doctor and his companions develop a counter-plot to kill all the Silurians and Sea Devils. Who will blink?  I won't spoil the ending (having outlined the rest of the plot!) but it is entirely in keeping with the rest of the story.

Overall, I enjoyed this series - it was initially difficult to get past the dated sets and costumes (especially the aliens), and the main problems I had was how an underwater sea base could be so bright, airy and spacious (even when it's under attack and the weapons system is disabled); and how the pace of the story was much slower than modern Doctor Who.  Being spread over 100 minutes meant that there was more time spent on each aspect of the story - there were fewer quick action scenes, and significantly less running around (in contrast to modern Who, where chasing and running up and down corridors has become almost cliched).

Next:  The Sea Devils (I get the feeling I should have watched this one first)