UKH

I attempt not to confuse kids for a living while increasing their maths knowledge in a school setting. This week bodmas/bidmas has led to kids being stressed, angry and properly swear at me across the classroom because it doesn't quite always work.

It's not a maths forum but I'm not a member of a maths forum so I post it here.

How many different examples of bodmas not working can you come up with and have you come across a better way of helping kids remember how to calculate expressions in the right order?

My aim is to convince our head of maths to ditch bodmas by lunchtime tomorrow.

Like when watching university challenge and I don't understand the question, I am going to guess....

... my answer is 6.7  

I don't understand the issue.  

Can you give me an example of when it doesn't work?  

What on earth were they trying to do that caused swearing? 

The only issue I can think of is the (mis)use of the divide sign when writing things out above and below a line would clarify things.  

FWIW I teach BIDMAS with well spaced printed questions and get them to use 1 colour to highlight or underline all the first bits to do.  When the questions get a bit more difficult you need 2 colours.  

I get them to write the answers to the first operation above the question and work through that way. 

Without knowing the age and stage I don't know if that is appropriate for you? 

Of methods there are many, of principles there are few. He who understands only methods is sure to have trouble. 

Teach 'em Reverse Polish first, that will make BODMAS seem common sense.

I'm not quite sure exactly what you mean but I think the main issue with the BODMAS rule is the divide sign. Above school level maths the divide sign isn't used at all because it's confusing, and fractions are universally preferred.

Just tell them when in doubt apply brackets and even when not in doubt think about doing so.

That's the only thing I can think of, and really that's not a Bodmas problem.  

I see what you mean, but in the context of teaching bidmas I suspect telling them to put more brackets in will cause chaos.

What was the problem?

Happy to send you what I use but it won't be until the weekend if that's any use?  

I've not got my hard drive or access to my emails until then.  

Bod was a shit cartoon and I never watched it at Xmas.

It pisses me the **** off.

Operator precedence is vastly easier to learn by just teaching it than by bolting on some stupid contrived mnemonic for which you also have to remember some stupid contrived interpretation.  

The best way to learn to do something is to do it.  When climbing do we use mnemonics for how to figure out a route, or do we learn to intuitively keep our balance whilst repeatedly moving the lowest limb up and re-finding balance etc?  We do this because climbing is a skill that has sweet F-A to do with the language part of the brain.

It's the same with maths - the language processing part of the brain has no f*****g place in evaluating expressions, it belongs in the visual/maths part of the brain.  So like with climbing, teach by doing starting with simple examples and raising the complexity.  This should be accompanied by explaining the process in words, but mnemonic recall and execution has no place in actually DOING it.  If you're teaching a kid to stop, do mnemonic recall and then interpret the mnemonics then think about the maths you're directly undermining the development of their maths brain components.

Also, think of the dyslexics.

If I did have to teach a recall based way of doing it, it would be a diagram of operator precedence - you want them to look at maths, see maths, think maths, not words.  Different brain parts.

()

X^Y

*/

+-

The one that always catches me out in Python programming is that the modulo operator '%' binds more tightly than any of the above, which leads to repeated confusing error message when doing string formatting with the unrelated '%' operator with some misplaced brackets.

I haven't heard of that stupid acronym since I was in primary school.

However, it doesn't even work when applied to 1 - 2 + 3, does it?

Surely it does. 2. Or am I missing something?

Nor do I.

I'd like one too.

The only issue with Bidmas/Bodmas is that teachers disagree with what the i or o stand for. Especially the o. I've no idea what the o is meant to stand for.

The BODMAS order says 'addition, then subtraction'. Which (stupidly) leads to -4 in the above.

None, so far. What had you in  mind?

swearing - i work with kids who have autism, and swearing happens often if they perceive something that we said that they've taken literally is then different when they try to apply it literally.

i.e. 

6-4+2 = 4 if you ignore a strict bodmas and apply sums together, both positive and negative, which is what we do

6-4+2 = 0 if you rigorously apply the order and do addition before subtraction

saying "you do d/m together and a/s together" doesn't help when they remember the word and that's supposed to help them remember an order.

So the student's comments were something like, "you've just told me 6-4+2 isn't zero but I've just read bodmas off the wall and you and all the other teachers keep using it but now you're saying it doesn't f**ing work!" And this wasn't an unruly rude student, just sheer anxiety at struggling to understand something.

I reckon it's a lot to do with crap questions in textbooks, removed from all context. If you were to create your own expression to model a situation, you'd be less likely to do things in the wrong order.

What's reverse polish?

Brackets, Order, Division, Multiplication, Addition, Subtraction

So, first we do the addition

1 - 2 +3 becomes 1 - (2 + 3) = 1 - 5

Then, the do the subtraction

(1 - 5) = -4

Which is totally wrong.   So, whilst the mnemonic implies an order, what it actually means is B > O > D,M > A,S where D and M are executed by the left-right order of the expression and not by their relative order in the mnemonic, and likewise A and S.

So, not only does it undermine the development of the maths part of the brain by pointlessly involving the language part of the brain in maths, it's a shit mnemonic that doesn't even work unless you remember which parts of it group together.

I always understood the AS part as addition and subtraction (left to right) not that one took precedence over the other.

I think the divide sign has a place when dividing a fraction by a fraction in order to avoid double decker fractions.

It always works. The only time it ‘doesn’t work’ is when the equation has been written ambiguously. So that should be part of the lesson to explain how important it is, not to fall into that trap when writing equations. 

The division sign is an abomination.  Teach them to use / and brackets to make explicit what is dividing what.   That is the real world solution.

I read a book in which the central point was that mathematical thinking and language processing were such similar things that it was silly to suggest that some people just can't grasp maths, when they've managed to learn a language.

His other point was absolutely that maths removed from context was difficult to teach and shouldn't be, in the same way that we use language as we're learning it, rather than learn all the language first, then try it out.

Damn. I hate having to agree with you!

Did you explain clearly to them in the first place that you do A and S in the order they come? I always write it with the D above the M and the A above the S until they are used to it.

Add and subtract are equal in the list.  

So are divide and multiply.  

I always make this clear. 

unfortunately 6 different teachers will have taught them to use it before us, we're secondary.

I'm a big fan of starting with context, then seeing what makes sense, instead of learning rules that you've got to remember. I'm just a teaching assistant watching others' trying to dig their way out of holes of their own and other staff creation and trying to stop kids falling in.

I really don't understand the problem. As I said above "I always understood the AS part as addition and subtraction (left to right) not that one took precedence over the other."

Have to agree with Marsbar and Robert Duran here. There's no priority for +- or for +÷. Just do it left to right... 

So, can I ask. How would you do it? Do you add before subtract? If so, I think the problem may be with you rather than BODMAS...

Was that "The Maths Gene"? The bit about the cards with letters on one side and numbers on the reverse which most Harvard undergraduates get wrong yet is logically exactly the same as rules about underage drinking which are easy to understand is brilliant - I always use it when teaching the contrapositive and converse.

However I think the main premise of the book that our ability to do maths is an accidental side effect of innate universal language structure is pretty controversial!

I could disagree quite strongly with that.  The lexicon of mathematics is far simpler than that of a spoken language such as English, but the set of rules for manipulating the lexicon is almost totally unrelated to that of a spoken language and the process of manipulation can be far more closely related to visual cognition than linguistic.  

I sort of agree - some people pick maths up for the sake of it, others need to use it to pick it up - and I don't think this is purely down to motivation but more about learning styles and a difficulty engaging with abstract concepts.

I would say - especially for autism spectrum students - that dressing up what could be quite simple concepts in endless descriptive language and contrived mnemonics that need extra interpretation adds extraneous context that makes their learning suck.  All that talking crap has nothing to do with the application and is confusing.  Where-as if you just walk the students through a bunch of examples and explain what you're doing, preferably with a simple diagram, the penny will drop.  I've had 1-on-1 interactions with perhaps  50 autism spectrum students teaching computer programming, including operator precedence, and walking them through an example and then explaining it in words is generally very effective.  

As an autistic person prone to over literal interpretations I can see why they didn't like it. 

You have to make it absolutely clear that it is just a way of remembering, and it contains groups.  

B

I

D and M

A and S

So for 6 - 4 + 2  all the operations are the same priority.  We just go left to right.  The answer is 4.  

If you had 6 + 3 x 2 then they are not all the same priority.  The multiplication comes first.

So I would expect to see working out above the 3 x 2 (only when learning, not forever) 

           6

6 + 3 x 2 = 12 

Yes, is often necessary to get kids out of bad Primary school habits!

What context do you use to explain why 2 + 3 x 4 should equal 14 rather than 20?

that's the book. So not a popular theory then.

Oceanrower21:41 Tue

In reply to Phil Lyon:

So, can I ask. How would you do it? Do you add before subtract? If so, I think the problem may be with you rather than BODMAS...

No, I'm forever stressing that +- take equal precedence so work your way along the expression. It's just that they seem to default back to simply remembering the mnemonic and then taking A then S as a strict order. This is my OP, that applying bodmas without remembering it's quirks doesn't work, so therefore doesn't help as much as it should.

I've got lots to chew on here from tonight, thanks all. 

There's no disagreement with that.  However, what it means is that BODMAS is really B>O>DM>AS.  So, if you're an autistic student who has naturally latched on to the idea of operator precedence and interprets it as B>O>D>M>A>S, it will be really frustrating to do that and to be told you did it wrong, and to have to remember additional wordy clarification of the acronym.

Cut the words out and draw a diagram - something like this, with a left-to-right arrow under */ and +- and tell them you start at the top, and for rows with more than one item you work left to right.  Adding the language / mnemonic overlay - especially with a shit mnemonic that needs further recall/interpretation beyond the mnemonic - is just a massive barrier to some students' learning.

()

^

*/

+-

I'm screaming inside.  Having a mnemonic drawn all over the place in 2D is going to make some autism spectrum and dyslexic students feel thick and all the words written on the wall won't help.  Just talking them through a bunch of examples and explaining it as it goes and perhaps drawing a precedence diagram of the actual operations - a world apart.

What do you think of SOHCAHTOA as a mnemonic for trigonometry then? The definitions are completely arbitrary, so no amount of understanding of concepts is going to help anyone know them. And almost all kids have no problem remembering the mnemonic and extracting the definitions from it when they need them.

I talk about when we're counting stuff how we normally collect things into groups first.

2 and 3 lots of 4 seems intuitive to interpret that way if we imagine objects.

Maybe 3 boxes of 12 eggs and 4 more eggs on their own. No-one would say that was 48 eggs, whichever way you thought about it.

Why are they learning BODMAS before they have learnt how to do basic addition and subtraction? 

Surely they have been taught 6-4+2 as a very early lesson in primary school. BODMAS must be secondary school level. 

Well I've read the entire thread and haven't a f***ing clue what it's about. 

No wonder kids are swearing!

I'm sorry to be rude, but frankly if I had a TA trying to give context to BIDMAS to autistic pupils while wanting to get it abolished because they didn't know that add and subtract are equal priority I'd be having a word with the Senco about sending me someone who can do basic maths without over complicating it. 

Autistic kids love stuff like BIDMAS when its properly explained.  Its predictable and it works. 

The only context it needs is that maths doesn't work if we all do different things, so a few hundred years ago people decided the rules and although it seems odd that multiplying is more important, that's the rules.  Generally it's the neurotypical kids that are interested in the story, the autistic ones don't care. (Sweeping generalisation there).  

I apologise for being blunt, but hope it saves you embarrassing yourself with the head of maths tomorrow. 

Edit

It seems you did know add and subtract are the same priority.  

Apologies.  However I don't think abolishing it is the answer.  Make the groups clearer.  

So how would you explain why 2 + 3 x 4 is 14 rather than 20?

And I really don't think kids are going to remember your precedence diagram!

Sine=Opp/Hypo. Cos=Adj/Hypo Tan=Opp/Adj

And that's 38 years after learning it and finding a use for it maybe once a year.

And I can still remember the formula for quadratic equations.

.

.

.

.

.

Mind you, I can't remember what a quadratic equation is...

I wondered if it might be this.

BIDMAS is useful, especially for weaker students who are going to get the order wrong without some sort of aide-memoire, mnemonic or otherwise.  I've always used it as BI(DM)(AS) as you need to emphasise that D and M are essentially the same operation, and so are A and S.  You also need to emphasise that in a long string of numbers and operators, the plus and minus always belong to the number immediately following them.  Sadly at primary these are often overlooked.

So have them think of 6-4+2 as being:

(6) and (-4) and (+2)

In other words, it's just the number line.  Start at 6, go down four places, and then up two.  It doesn't matter which of them you do first.

But 6x4-2 is

(6x4) and (-2)

which is where a method of remembering that (DM) comes before (AS) is needed.

You may well get some return from showing them how BIDMAS is actually just a way to memorise the operators, as Wintertree had it above.  This has really useful consequences, because they're about to enter a period of their education where they will repeatedly find (in many subjects) that what they were taught a few years back was a gross simplification and here is a new, more complex and better way to think about it.

Adding the language / mnemonic overlay - especially with a shit mnemonic that needs further recall/interpretation beyond the mnemonic - is just a massive barrier to some students' learning.

I'm hearing this loud and clear, thanks.

Yes they have. What has to be avoided is them starting to get it wrong when they come acfrodd BIDMAS!

It's a much simpler mnemonic to execute.  It breaks down into triplets SOH, CAH, TOA where each triplet is of the form FUNCTION(ANGLE) = SIDE / SIDE.  To use it the student has to memorise it and memorise two consistent rules for interpreting it - rule 1, break into triplets, rule two interpret as function(angle)=side/side. 

That is absolutely not the case for BODMAS.  So SOHCAHTOA  isn't broken as a teaching device for dyslexic and autism spectrum students in the way that BODMAS is.  Having said that, I don't remember it.  I gave up on it and learnt the functions by tapping sides on a drawn triangle and associating tap patterns with operators.  To this day I do a little mental picture and tap in my head for sin and cos.  I remember tan as sin / cos.

Totally agree.

But it might be a special promotion where each box has four extra eggs for free.

I'm not convinced.

Unless you start writing a lot of maths as computer code... 

Sorry, I've not made it clear enough that I am absolutely fine with this. They're equal priority. And I reinforce that. And I still suggest that the bodmas mnemonic, half-remembered, implies a preference for addition before subtraction. 

Jamie Wakeham, you're making a lot of sense there.

If we had to have an acronym I'd choose BEPS (Brackets, Exponents, Products, Sums)

I make mine draw their own version with both symbols and words.  The important thing is to group the DM and AS somehow in the picture.  

Some kids might remember the symbols as wintertree draws it.  Not that many of them, but I suspect more with autistic kids than mainstream.  

I think you'd just get chaos from calling division a product and subtraction a sum. 

As I said - I would take them through examples, explaining the order for each one, to develop the understanding in the maths part of their brain first and to back it with understanding in the linguistic part.  I wouldn't start with the linguistic part.

Most will remember BODMAS just fine.  A few wont and for those, it's worth trying a diagram that uses the same symbology as the maths and lays it out visually not linguistically.  I am firmly of the view that dyslexic is not a disability (except for how teaching and workplaces are currently set up) but a different and often more visual way of approaching things.  One that screams on the inside when confronted with too many words.

This is useful, but I actually think that far from being arbitary, multiplying first does make sense. I guess this whole thread, if I started it again, would be asking for ideas about showing this better than the crappy eggs example I gave.

Totally agree, I'm not dyslexic but I am a very visual person. 

confusion about division being a product probably,

although at some point in secondary education it's really useful to think of division as a product of it's reciprocal. In fact, they do that in primary anyway don't they, what's half of that cake? Same as the cake divided by 2. 

If you were in mainstream, this might be interesting.  Trust me, it's of little interest to me and likely the same for autistic children.  We don't actually care why for something like that.  

Yes, but which part of the maths brain UNDERSTSANDS why 2 + 3 x 4 is 14 rather than 20?

Fine, so lets not reject BIDMAS  if it completely if it works fine for the vast majority.  If it is no good for those on the autistic spectrum then obviously they can just remember the symbols - I don't see the problem.

Which works nicely if you teach that division is multiplication by the reciprocal, and subtraction is addition of the unary negation.  This is the way I think about maths - and why I can't get worked up over a division symbol vs a two-line fraction, but unary negation isn't taught as a separate thing at school.

It does.  "The rule" is that precedence is given by the acronym.  Except where it isn't, when some other rule applies.  Which isn't indicated by the acronym.  Simples.

ok, thanks.

I need to remember that we all have preferred learning styles. I hate not understanding why and how something is a certain way and just being told to learn a rule and use it. 

I now realise many people cope just fine with that.

That's true. Sometimes when I'm writing C or similar I end up doing something like

c=a/b/2;

and I feel horrible inside and have to rewrite it as

c=0.5f*a/b;

which in many ways is uglier.

Just stick with BODMAS and explain it doesn’t change any maths they’ve already learned. It only comes into play when they learn about Brackets and Indices anyway. I suspect you’re introducing it far too early. 

hence the swears

hence this thread

I don't think I'll change the world tomorrow.

It'll be fine.

If you can't teach them that addition doesn't take precedence over subtraction, you're absolutely screwed trying to get that into them!

Good luck trying that approach with a typical bunch of 12 year olds. 

The same part that can look at 3 items and know that there are 3 without counting them.  It just learns to do it.  

When I look at the maths you wrote, my eyes scan it all, however over the (3x4), compute that, then they go left a bit and do the next bit.  I don't "think" about it, the bit of my brain that "understands" and can explain it doesn't seem to be involved. I just do it.  I've learnt it as a reflexive skill just like climbing.  Somewhere in my brain, a part takes over my eyes and uses the internal visual abilities to do it.   There's no linguistic thought going on.  I kind of assume that's how everyone does maths...

I don't believe I've called for that.  I've said why I find it shit, and given my interpretation, and given some suggestion of an alternative for people who struggle like I did with it.   

If they - or especially dyslexic students - are part of a class being taught BODMAS - perhaps they don't get it, they struggle and they feel like they're swimming in treacle whilst everyone else "gets it", they feel thick and vulnerable and disengaged and miserable and like a failure.  So that's the problem.  

Sometimes the rules are just a certain way because someone decided that’s the way they’d be. 
 

Traingle ABC has sides a,b and c. Where a=BC and is opposite angle A, b=AC and c=AB. The vertices are labelled anti-clockwise.  It’s just convention. No reason why a couldn’t have been AB and the vertices clockwise. 

That is why I use the underlining and colours, it gives a system which helps the one that are struggling. 

I suspect your lack of language when doing maths isn't what everyone does, but who knows?  

Yes, but you must have, at some point in the distant past, been told to do the 3x4 first. I agree that it becomes reflexive after enough practice, but I think kids need something to fall back on until it has become reflexive (and quite a few will not do enough practice for it to ever become so). 

I noted that one of the key concepts isn't taught in schools, so I think it's pretty clear I wasn't proposing this as a solution.  Just commenting on my view of things.  Division as a reciprocal to multiplication is possible however. 

I think the point is being missed - the kids in question likely understand the equal precedence of addition and subtraction, but then when they are taught that an acronym gives precedence and because a non-monotonic precedence sequence is forced into a monotonic acronym they over-literally interpret the acronym and can't square the two off and get confused.

I wouldn't expect you to "understand" that because your brains don't think that way.  Having learnt it easily long ago and taught it for many years, it's simple and obvious.  The important thing is to take other people's word for it that not everyone - including a few of those that you may go on to teach - sees the world the same way that you do or learns in anything like the same way.  

Robert, I know you teach in a school so you perhaps meet ~30 new children a year?  Teaching in a university I got a new cohort of ~300 students every year, and I read through all their paperwork on learning support and I meet with quite a few autism spectrum and dyslexic students 1-to-1 to unpick their problems.  I've probably met and taught more students with dyslexia and various spectrum disorders in the last 5 years than a school teacher might in a career (as became embarrassingly obvious when asking a head about their dyslexia support one of the smaller schools we visited looking for a school for Jr.)  I've seen adults reduced to tears over their inability to understand something that 95% of the class gets straight away, and one of the absolute most rewarding parts of my job was to unpick what their barrier to learning was and to explain it in a different way to help them get it.  

Are they? Never come across that!

Except that it would offend any sense of symmetry!

I much prefer that to a/b/2 which I find visually offensive and just weird.

Sorry.... although I would have written a / b / 2.0f before feeling bad and re-writing it as 0.5f * a / b; with some spaces...

Yes - but the point is I learnt by doing and having explained, and not by mnemonics.  

I was upset when the stupid film title said it was East Of Java! 

That's the important point - not to have an over-reliance on one system so that different brains all have a chance of having something that works.

I suddenly, really really want to set up a research project studying how people solve maths in their head, reading it off a monitor and using an eye tracker to study their gaze over the maths.  Doing that with dyslexics and a control group would be absolutely fascinating.  It's been a decade since I did anything with eye trackers but I could be tempted back.  It would need a good psychologist to lead it however.  There must be some prior work out there - tomorrow's reading...

Surely you mean

Sir Oliver's horse came ambling home to Oliver's aunt?

I suspect I "store" some of my maths in the same bit of my brain as my motor skills, my first instinct would be to write it or draw it.  

The only thing I know about eye movements is that we were told that putting key words on the very top left might be helpful, something to do with eye movements when learning spelling.  No idea if that is myth or science.  

is that for real?

Six old horses, cold and hungry, trampled over Albert.

Or 

Sex On Holidays Can Always Help Teachers Overcome Anxiety

Motor/visual - they’re very linked.  You can visualise muscle movement without doing it and often my finger does an imaginary dash around the maths as I execute it.

With eye movements I was thinking does someone read the maths left to right, or do they first pick out operators and brackets?  My first conscious impression of an expression is of its structure and that impression contains a “depth” based on the operator precedence.  I’m guessing my eyes dart to the brackets and operators first - they’re visually very different to the numbers which helps, as does the visual symmetry in brackets.

I think it is important to make the distinction between different ways of coming to an understanding of something and different ways of remembering something (though obviously something is more likely to be  remembered if it is understood). The trouble with arbitrary rules is that there is nothing to understand. The problem with teaching maths is that we so often expect kids to remember stuff they are not yet ready to understand, so it effectively gets reduced for them to arbitrary rules.

Don't anyone get me started on the CAST diagram.

Good.  The process of drawing something I think is a powerful way of putting stuff in to long term brain storage way avoids passive listening.  There’s something almost magical about the ears>brain>pen process, even if you then bin the paper you wrote on.  When lecturing I laboured the point of drawing diagrams with me, even though they were all in the companion notes.  The process of drawing the precedence chart with the symbols will probably help the distressed student the OP posted about way more than layering more words on to a shit mnemonic.

You keep saying it's a shit mnemonic.

It can't be that bad if I can remember (and understand) it about 45 years after being taught it!

I don't think I'm thick. I had a pretty standard secondary education and got 9 of the old fashioned 'O' levels and 6 cse (and an ungraded in Latin...) followed by a couple of shoddy degrees but

"Which works nicely if you teach that division is multiplication by the reciprocal, and subtraction is addition of the unary negation."

Give me BODMAS any day. I don't even know what a unary negation is!

I'm pretty sure that is what most people do with experience, but I'm also pretty sure it only comes with practice having been taught to do so. Just like many motor skills need to be taught and practised before they feel automatic.

I take your point, but there are two things to understand about arbitrary such rules.  The first is that they are arbitrary, and the second is that they exist for a reason, which is to have a standard form of communication - with ourselves and with others.  Arguably there is no deep understanding to these rules, only learning to apply them.  Most take to that naturally, some who look for order and reason will be distressed by the arbitrary nature, and for those few explaining why we have crystallised an arbitrary set of rules can help them move on from the distress.  It’s a hard thing to explain if you haven’t experienced it, and most people never experience it.

The time was, some on UKC would argue that operator precedence wasn’t arbitrary but had some logical origin...

Unless it's between the 15-year-olds on a residential

Or a 15-year-old and a teacher

 

So true.  

If you use a stupid mnemonic, which encourages them to rigidly evaluate in strict order, from L to R, then the mnemonic is the problem. I can see this especially being a problem with the rigid thinking that can come with autism.

It is.  If it was B>O>DM>AS, it wouldn’t be shit but it wouldn’t be a mnemonic. 

You miss the point.  It worked for you.  It works for many people.  It didn’t work for the distressed student the OP posted about.  It probably doesn’t work for about 5% of pupils who are otherwise perfectly able to get the concepts.  

Fundamentally as a mnemonic it *is* shit because it doesn’t encapsulate all the information it pretends to.  Most people can deal with that - most of our lives are spent dealing with sub optimal systems - but for a small minority it’s toxic.  

Nor I hope do you think I have suggested otherwise.  I did not intend to.

I doubt many people do, even practicing scientists.

1 - 2 # subtraction of two from 1 (binary, two numbers)

-2 # The number “minus 2” (unary, one number)

1 + (-2) 

The minus in the first line is very different to that in the other two.  The difference is normally academic but becomes very real when you makes a rules based system to execute written mathematics (for example a computer language compiler).

That's why I go top to bottom and put the same ones on the same line.  

And, anyway, I don't think BODMAS even IS a mnemonic.

At the very best it's an initialism...

Well, due to the “o” in particular it isn’t an acronym:  a contrived reduction of nouns yielding mnemonics.

I try not to think about initialisms lest my SUVAT rage manifest itself.

Some Officers Have Curly Auburn Hair To Offer Attraction.

CAST: SouthAll Technical College...

A mnemonic is merely something to aid memory. It could be a phrase. Or an initialism. Or a diagram.

Exactly my point from my first post - https://www.ukhillwalking.com/forums/off_belay/bodmas_made_a_kid_swear_at_me_t...

Motor skills are taught by *doing*, not by acronyms.  It’s akin to muscle memory but for the maths brain.  Muscle memory comes from repetition and doing, not from acronyms.  Work through a set of exercises designed to train the “maths-motor” skills to be automatic and explain as you go so people understand why they’re doing it.  At no point does this need an acronym any more than learning to belay does.  

I learnt to belay through doing it, and through having it explained to me what each point of the motions was doing.  Practice, explanation.  No mnemonic that didn’t even encapsulate absolutely key details.

This causes a lot of confusion for later on I feel - when sides and angles are labelled differently. I used to teach based on opposite angles, angles between two sides etc to avoid this..

And for complete clarity, this was teaching maths to physical sciences students at a red brick university, who still struggled with basic algebra and concepts.

I don’t particularly like the tone of some people in this thread who seem to think that OP has inferior maths skills because students become frustrated with maths or aren’t receptive to particular methods. I can’t think of a day teaching when someone hasn’t been frustrated!

In fact, feeling frustrated for several hours and then finally figuring out the answer is one of the reasons why I enjoyed maths in the first place. It’s just getting students to appreciate this  

SATC?  FFS. That's even worse than CAST. It's not even cyclically in the right order.

Yes, but unfortunately in the real world, there simply isn't time to work through enough examples for it to be automatic for many pupils - there is always a new topic to be moved on to and a syllabus to get through in time for an exam. This is why I sometimes hate teaching maths and just want to give up - we try to teach far too much in the time available so that there is no chance of either fluency or understanding for many pupils, so it inevitably gets reduced to meaningless memory games. I hate it.

Totally agree.  The critical thing to me is that the OP recognised a source of grief and sought other views.  That’s easy for me as I worked in a building with 50 other people teaching basically the same subject.  For a teacher in a school there are perhaps 0-2 other people they can talk to, so I’m really pleased to see the OP posting here.

Yes, I can explain the reason why we do each part of the belaying action, but you still havn't explained to me why 2 +3 x 4 is 14 rather than 20 (at least in a way intelligible to the vast majority of 12 year olds).

If it's pronounced as a word ('bodmass'), then it's an acronym. If it's read out as letters (B-O-D-M-A-S), it's an initialism. FBI is an initialism. NASA is an acronym.

Just to be difficult, ASAP can be both acronym and initialism, depending on how you say it (aysap or A-S-A-P).

Oh, and it's not just contracted nouns: laser, for example.

It's not done cyclically. Why would it have to be done cyclically?

Four quadrants, top left, to bottom right. Overlay those words in two lines over the quadrants, just as we write lines of text.

South            All

Technical.    College

CAST is especially contrived, because it starts in the fourth quadrant, rather than the first quadrant (0-90°).

I have explained it several times, but I have not tried to explain it for 12 year olds.  I’ll give it a go

• We get different answers depending on if we do the + or the × first 
• Maths is only useful if everyone gets the same answers - if everyone agrees.  We all have to speak the same language
• Everyone does the × before the +; so that is what you are taught to do, so that you agree with everyone else
• To be totally clear and unambiguous we could write 2 + (3 × 4) but many people are lazy and rely on everyone agreeing that × comes before + and so don’t write the brackets.

Because angles are by their very nature cyclic.

At least it is cyclic. SATC is neither cyclic nor starts in the first quadrant. I presume it is used because Cast is a word.

The only logical one is ASTC. I refuse to use anything else and explain to pupils why.

All science teachers are clever (or crap).

OK. That is more or less exactly what I tell 12 year olds. But it only explains the point of convention, not why we choose this particular convention. In the end we are simply telling kids to remember an effectively arbitrary rule - it is not a situation where understanding will help memory.

So what? It's a mnemonic. It worked. The mnemonic isn't SATC; it's South All Technical College.

Curly Auburn Hair has nothing to do with angles, but it works as a mnemonic.

Bye Bye Rosie, Off You Go, Birmingham Via Great Western has nothing to do with numbers or colours, but it works as a mnemonic for remembering resistor colour codes.

Of course, if you can remember what the sine and cosine functions do, you don't need any of those mnemonics; just remember what the wiggly lines do. The wiggly lines are easy to remember. If you need to remember which starts at zero, and which at one, just remember what each represents (opposite or adjacent).

Well it shouldn't work! It's an affront to what is actually going on.  

Which bit of mnemonic are you struggling with...?

It was used by my o-level advanced maths teacher. He had come from Southall... It was instantly mnemonic. It was simple. It was easy to remember. It serves its purpose perfectly.

Sorry, it just offends me that it is in the wrong order.

You can just remember the wiggly lines, but the ASTC diagram is more fundamental - to understand the wiggly lines, you need to understand the ASTC diagram and calling it an SATC diagram undermines what it represents.

hsilop?

I'm with Captain Paranoia on this one, with a potentially more offensive anecdote.

I remember being taught CAST, and finding it hard and seemingly arbitrary that I had to remember which quadrant to start in and whether to go clockwise or anti clockwise when labelling them.

Fortunately, the lad who sat in front of me and my friend had the initials SA, and we could generate all sorts of puerile initialisms about him and use that to remember the initials in reading order SATC instead of CAST.  I jotted "Sam A's Titanic Cock" in my notebook and passed it to my friend.  We got a very stern word from the teacher for uncontrollable giggles through the rest of the lesson, but that's the memory aid that stuck and helped me until I could memorize the wiggly lines.

Sam's phallus continued to help me check answers in my mech eng degree and still occasionally now in industry.

Strictly applying BID/BODMAS similarly legged me up earlier in school until I became more fluent in interpreting written maths.

I've never managed to learn all the exceptions/not-exceptions to "I before E except after C" and mess it up on a daily basis.

There is rarely one 'right' way of remembering things.

I'm from an era of trig tables. The primary purpose of that mnemonic was to allow those (first quadrant) tables to be used over all four quadrants.

I'm not generally a fan of mnemonics, as I often struggle to remember what they mean, but this worked for me. Now, I would try to use the fundamental four quadrant diagram (I would NOT call it an ASTC diagram), with a vector at the appropriate angle, and show how it resolves into sine and cosine, in which case, the sign can be read directly.

But for the "CAST" diagram to be of any use you need to know which quadrant is the first one and that they go anti-clockwise anyway! It just seems crazy to have a mnemonic which ignores this.

Isn't the answer here to teach pupils to use brackets, rather than focus on artificially ambiguous expressions?

I don't like mnemonics either. They are usually just there to replace understanding when there is a maths exam to be passed, which is pretty depressing. I don't recall coming across Sohcahtoa, Cast or Bidmas until I actually started teaching maths, presumably because I must have been good enough and keen enough on maths at school to have replaced them almost immediately with understanding or practice (if I was ever told them at all). I also didn't learn my tables until I started teaching maths - I think I realised at a very young age that I only needed to know the products of prime numbers, so never bothered learning my tables.

It sounds like the missing principle is that the  × is shorthand for multiple additions, i.e. 2 + 3 × 4 = 2 + 3 + 3 + 3 + 3

Once that's understood, the necessity of an order is obvious 

Why shouldn't it mean 2 + 3 + 2 + 3 + 2 + 3 + 2 + 3 ?

So never write 3 + 2a ? Algebra would be rather cumbersome. Having said that, the omission of the x sign when writing algebra makes the order seem natural. Maybe we should ban the x sign and just use a dot right from the start.

How sad - and somewhat alarming.  

So off I go to learn how to teach.  Get to a school and the head teacher insists that I use certain mnemonic to teach maths?

Some mnemonics make it more complicated than the original problem.

Because it doesn't, you've just added a set of imaginary brackets to support your argument. The rules of the language are unambiguous - unless people intentionally produce confusion like that. 

Of course the easiest way would be for us all to use brackets all the time, then then would be no opportunity for misunderstanding the rule that's set out.

And you added a set of imaginary brackets to support yours. Without the arbitrary convention it is ambiguous.

When you use a cloth or applicator brush initially when cleaning your boots to the required standard you are adding polish to the boots.  Reverse polish is when you are getting a high standard of shine using the second brush to remove Polish and gain a shine, subtracting polish

I have to stand up for the usefulness of SOHCAHTOA in life - it has stayed with me right trhough life to the present and as someone who is not a maths teacher but has had occasionally to use the information contained in said mnemonic it has been very useful, even to the point of surprising much younger colleagues who could remember no trigonometry at all.

I also remember ERCATLI and Richard of York Gave Battle In Vain much less usefully.

Mnemonics can be good for recalling old knowledge

the only bit of the tables I could remember was 6 x 7 =42 from which all the difficult bits I could never remember could be obtained and on the subject of tables I think it was the use of 4 figure tables that defeated me at o level, though I managed  a middle grade pass having done only about 3/4 of the paper because of slowness of carrying out operations using paper tables

For me the easy way to work it out is to treat the -2 as one whole number, not an action on the 2, if that makes any sense?

so 1-2+3 becomes 1+-2+3 or 1+(-2)+3

No no no, he means

Silly Old Henry

Caught A Herring

Trawling Off America

It all works very easily for addition and subtraction if you just imagine a calculator with only an accumulator and no memories.

That forces you to compute any other result externally and reinsert it into the sequence in advance of addition or subtraction.

I still use FOIL for multiplying out contents of brackets and that was taught me at about 9 or 10 iirc (as well as making a face out of it as you do it which shows you've done it all)

It functions consistently with how it is applied in more advanced maths, it's just that that is written in a stupid format. If you want to reorder an expression you can force it using brackets:

(1-2)+3=2 (Resolve the expression in the brackets, then do addition)

1-2+3=-4 (Resolve the addition first, then do subtraction)

1-(2+3)=-4 (Resolve brackets first, then do subtraction)

If I saw 1-2+3, or an algebraic version of it, in a report I was checking I would query it and request that the calculation was broken down into multiple steps.

I got by without learning which was the first one, which goes some way to refuting that you "need" to "know" it.  I knew the A stood for "All" so that's where you start with all of them, in the upper right hand side.  By drawing a vector in that quadrant, and knowing that its the angle between it and the horizontal that we're talking about, I can remind myself each time that you go anticlockwise from over there on the right.  Sounds similar to Captain Paranoia.

That might seem more arbitraty and convoluted, but it helped me solve the problems.  Not just to pass exams - I still know all of that and fetch it all back from the depths of my memory when needed to solve "real life" problems.

Which I think was the point of the original post.   Certain mnemonics don't work for some students and alternatives are needed.  To say they replace understanding is a bit brash.  They can be a framework for helping the concepts to land, or a shortcut or reminder for how to apply them.  But how frustrating to have to teach one method!

I know that trig deals with three ratios, one is the ratio of the opposite to the hypotenuse, one is the ratio between the adjacent and hypotenuse, and one is the opposite and the adjacent.  By knowing that, I can derive the squiggly diagrams.  sohcahtoa helps me remember which of the nouns "sine" "cosine" and "tangent" apply to which ratio, and check that I have the ratios the right way up.

If only we could reverse polish!

Let me assure you, the head doesn't give a monkeys.  They're far too busy!

However, every maths teacher these pupils have ever had will have used BIDMAS or similar.  Realistically we aren't going to remove it from education.  The best we can do is to make it work as best possible.

There is a benefit from everyone using the same system, of course.  I know I can walk into any classroom in the country (or, these days, take on a tutee from any school) and use BIDMAS. The most confusion possible is that they look at me funny and say 'surely you mean BODMAS?'  It's just like I know I can use SohCahToa or ROYGBIV - they're universal.

Ive put it to the department this morning, will feedback responses 

The problem with acronyms is that they can get in the way.

As a student learning the Kreb's cycle, we were taught the structures and how they changed at each step. Then someone told me the rather non-PC acronym 'Can I Kiss Sexy Susan Far More Often' and that pushed everything else out of my memory cells.

Mmm.  I think I disagree.  It's important for us to keep in mind when teaching maths that we are, almost certainly, the best mathematician in the room.  We can't assume that all our students will have the same ability to see why things work, or even the inclination to bother to learn it.  A lot of them just want an easy way to get this bloody question right so they can move on.

At a guess, I'd say that much less than half the pupils I've ever worked with will have had any idea at all of why SohCahToa works.  Will I teach the brighter students why it works?  Of course.  But the fact remains that if you get lower sets parroting SohCahToa then they'll get those trig questions mostly right, and if you try to teach them how to derive them from first principles then they'll get them mostly wrong.  

This is even more true for arbitrary lists, such as ROYGBIV (along with Blue Bends Best, which is a damn sight easier than recalling which wavelengths have a slightly higher refraction in glass...).  There's no particular reason you should suspect that red wavelengths are the shortest or that green comes before blue.

Are we replacing meaningful learning with pointless repetition of memorised facts?  Maybe...

Until last night I’d never heard of CAST and I did engineering mathematics at university.
 

The pictogram is just as easy to remember for me. 

I suspect we use the same mnemonic for resistor colour codes.  I was taught it by my grandfather, who was a Navy submarine electrician.  I happily trotted it out in a DT lesson at the age of 11... it didn't go down too well.  Once or twice I've tried to overwrite it with a less racist and misogynistic one but it's so firmly stuck I just can't get rid of it.

Could you teach that subtraction is adding a negative. Then you just have BODMA and don’t need to worry about subtraction at all?

I agree - it's always important with an arbitrary rule to explain that it's arbitrary and yet important to shared understanding.  Sign conventions are another example - way too many Physics graduates go through life convinced an electric field actually points in the direction the arrows are drawn in and don't appreciate how the sign convention is arbitrary and how it cancels out of the final result....

I think there is a basis for the operator precedence that is logical - consider the change in order of magnitude between the two inputs to a binary operand and the output - addition and subtraction don't change the magnitude (in general), multiplication and division do, raising to the power does far more so.  Brackets by their nature trump all.  For the minority of students who struggle to learn something arbitrary without a logical reason behind it, perhaps a description tailored from that will help. 

Quite. Subtraction is just adding a negative number and division is as has been said simply multiplying by the reciprocal. 
 

I think what’s being missed here is that introduction of a mnemonic before the principles behind it have been explained. Or rather using BODMAS to explain the order. Rather than explaining the order and then - if you have trouble remembering what order to do the 4 operations in (there’s only 4 to remember!) - here’s a handy aide memoir. 
 

It’s like teaching someone Roy G. Biv and then explaining this is used to remember the colours of the rainbow which are Red...

Its being done arse about face. 

I tend to teach them that adding and subtracting are a pair of inverse operations, likewise multiplying and dividing.  

This explains why they are paired up for BIDMAS and prepares them for using inverse operations in algebra.  

I think if I tried bringing in negative numbers at this stage it would cause confusion and upset.  But I mainly teach those who struggle so it may work for others.  

I don't think resolving brackets works in the way you think

1-2+3 resolve addition first = 4-2 then resolve subtraction = 2

1-2+3 resolve subtraction first = -1+3 then resolve addition = 2

if you add brackets

(1-2)+3 then this is 1(1-2)+3 = 1-2+3 = 2

1-(2+3) then this is 1-1(2+3) = 1-2+3 = 2 [edit 1-1(2+3) = 1-2-3 =4]

Negative numbers are taught very early on. I remember my children doing number lines and learning negative means going to the left and positive to the right. 
 

It’s almost as if someone is bringing in BODMAS and overwriting everything they have previously learned. 

That’s wrong. 
 

Addition and subtraction are done left to right. The brackets just alter the left to right precidence. Hence then B in BODMAS. 
 

1-(2+3) = -4

I don't start with BIDMAS.

I start by giving them a question and a picture of 2 children with different answers and ask who is right? 

Depending on the class we might work out how each got the answer, we might also use a scientific calculator and a cheap calculator and get 2 different answers.  

This way of introducing it shows that we need rules or it will all go wrong.

We then go through the rules. 

Then we use BIDMAS to help us remember the rules.  

Then we do some examples and practice.  

Sure; you can have negation and reciprocation acting on single numbers, and then you negate and add and or reciprocate and multiply to subtract and divide.  Then you have BOMA with M and A each splitting two ways.  But this is not how maths is taught at any age in school and so it would just confuse the hell out of most people.  

Never teach someone how to do something wrongly, even as an example. That’s how confusion sets in. The human brain doesn’t work that way to discriminate when learning new ideas. 
 

Just teach the right way. And if someone gets something wrong then you correct them. 

I'm usually teaching kids that struggle with maths.  I'm not "over writing" anything and I may choose to use the negative number thing with certain kids.  However for the majority its easier to not overload them with pulling together all the concepts at once.  

I'm just looking at it another way.  

If you take a number and multiply by 3 then to get back to the original number you divide by 3.  

If you have a number and subtract 2, to get back to your original number you add 2.  

It's a very very easy concept and useful.  

As for negative means you go to the left, it doesn't. That's exactly the kind of thing that gets mis remembered and causes utter chaos.  

Adding a negative goes to the left.  Subtract a negative doesn't.  Nor does multiplying (assuming integers)

so is 1-(2+3) not 1-1(2+3)?

edit ahhh

1-1(2+3) = 1-2-3 = -4

When you've got 20 years in the classroom come back and tell me how to suck eggs. 

It's called learning from mistakes.  

Or common misconceptions.  

Knowing what went wrong stops you from making other people's mistakes.

Correcting other people's mistakes (even imaginary people) is far more likely to increase self esteem in kids that have been told they are no good at maths. 

Me correcting their mistakes just reinforces their feeling of being bad at maths.  

Yes. But it’s not 1-2+3, it’s 1-2-3 = -4

2+3 is 5

-1 × 5 = -5

1 - 5  = -4

Making mistakes in the classroom (or under controlled conditions) is how you learn. You shouldn’t be showing incorrect methods as examples until you’ve taught the correct methods. 
 

So you explain how brackets work first. Explain how they change the order of the sums and show how if you didn’t use the brackets the sum would give you a different answer. Maybe that’s how you do it?

But we never give examples of the wrong answer or demonstrate incorrect methods until we are sure everyone understands and has learned the correct way first. 

I don't know why this is disliked, its valid. The +3 becomes -3 when  you multiply out the bracket.  

You do what works for you. 

I will stick with what works for me. 

Do you work for Ofsted or something?  

It isn't an example.  Except an example of why you shouldn't trust a cheap calculator perhaps, or an example of why we need rules in the first place. 

It is a question.  It is to make the children think.  I appreciate this isn't fashionable in these days of exam factory schools, but there you have it. 

Depends if you expand the brackets first. 
 

-1(2+3) would imply expand the brackets to me because it’s in the form -x(y+z) = -xy-xz 

It gives the same answer but depends whether you’re teaching it as BODMAS or an introduction to algebra. And in this case it looks like an algebra equation. If you put the x symbol in then I’d go for BODMAS. 
 

Weird. 

not only that, but in the years since O level I know far more about trig than I did at school - experimenting with BASIC and plotting using a home built computer in the early 80s showed me far more than i learned at school and SOHCAHTOA became clear   To the extent of understianding how the Ordnance Survey mapped the UK from forst principles.  sense at last!  Incidentally it is only in the last few years that I taught myself how logarithms work rather than just having to use tables.  I am 64.

Never had BODMAS or BIDMAS at school - the first I heard of priority of operations was when I read

a FORTRAN book and then learned BASIC.  But I was a very proficient early reader but completely unaware that anything like an alphabet order of letters existed till I was about 9 so it might just have passed me by - my teachers all wrote that I was dreamy or in a world of my own in class.

I applied the squared paper in the arithmetic book at primary school as an aid to drawing nice symmetrical suits of armour which was far more interesting to a reader.

No. We do electrical training. It’s quite important that people learn to do it right first time. You don’t get to rub out your mistakes. 

Thanks for making me not think I was going loopy.

I would hope I can do basic arithmetic given that I am about to submit my application to be a chartered engineer!

So you'd never show your students a terminal block with a loose wire and some scorch marks and ask them what happened?  

How do you teach fault finding?   

I don't remember being taught Bodmas either.  I wonder if we were.  I expect we must have been.  

we had a hut with old radios at school - I threw myself across the floor fault finding when about 16 when I felt the 600 volts of the transmitter supply of an old WW2 tank radio after testing with a screwdriver.  I was alone at the time ...

Did you do it again? 

No!  Well I did the fiddling, but with far more care!

I honestly don't know.

I'm just glad I seem to have gone through education, various jobs involving mathematics and have almost reached retirement without having ever heard of BODIMAS or whatever the hell it's called. Or the weird way they do long division these days.

I am officially a dinosaur, and a thick one at that.

Lesson learnt!  

I've certainly had classroom situations where I've felt that the application of 600V would help matters...

You don’t start with showing bad installations. 
 

You start by showing how to do something correctly. You let them make off and terminate. At that point they’ll start making mistakes. Then when they’ve learned they’re shown the effects of big problems. 
 

I don’t think a scorched terminal is a good example. You can see what’s wrong instantly. 
 

If you look at it from a climbing point of view, you’d teach a correct figure of 8. You’d explain he the figure of 8 works. You wouldn’t show them granny knots failing etc. You’d show them a fig 8 with appropriate tails as you’d explain why the tails are the length they are, but you wouldn’t tie a long tail and short tail fig. 8 to demonstrate. People learn visually and they’d then learn three versions of fig 8 which isn’t best practice. You’d then let them tie their own ones and correct the short and long ones. 

I never learned BODMAS just learned the order. Wasn’t until I saw one of those ambiguous trick questions on Facebook that I had to go look up what everyone kept saying. 

Everywhere I have worked has had brackets as part of the software development standards. It removes any ambiguity when you are looking at someones code a few years later plus some languages use alternate approaches (generally a simple left to right).

especially when the input voltage was 12 volts from huge lead acid batteries  - appliance of science to produce a startling stimulus to learning!

Actually it would be interesting to assess the risks in that environment, electrical, chemical, heights (I once locked the key in the hut and had to break in through the skylight to save face, not to mention poking people in the eyes with pointy aerials, or 30 foot telescoping masts collapsing or telephone cables being laid across a public road by pupils!

wow, what a lot of ideas, thanks

In looking through practical problems we've set and also past gcse exam papers, the only time you seem to use bidmas in education is to answer the questions that are deliberately set ambiguously to check you know your bodmas.

I struggle to understand the need for BODMAS.

In real life if you take the time to understand what you are calculating then the order of the operations becomes apparent.  If you don't understand what you are calculating then BODMAS is not going to help you and you probably shouldn't be trusted with the calculation.

Mondite beat me to it.  With software I use brackets as it reduces ambiguity and makes the same visual pattern for every operation.  It's a useful way of groping things together in a written environment that doesn't support multi-line fractions for example.  It's similar to how I always use curly braces for a C conditional be they required or not, as reliance on the rules for implicit scope just introduces space for confusion.  I believe MISRA C has an advisory rule for this for example, 5.something - but my copy is buried in some moving boxes from emptying my office and I've had no burning desire to dig it out...

http://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html

I got 18/5 for assuming a missed bracket in the divide.  

Shows why we should use a horizontal line not a slash for division. 

I think that is a divide issue not a BIDMAS issue. 

Thank you , very interesting.  I really enjoyed the challenge of maths in my 1st years at  school, but I cannot remember us using any mnemonics to remember anything - but that was in 1961-1966.

Except perhaps the colours of the rainbow.  Rich Of York.........   etc., But I can't remember the mnemonic any more and I 'm not sure I remember all the colours either - but I know where to find them *(google!)

ditto

were we both wrong 11 or 2 is the supposed answer!

Faced with that equation, with the absence of any brackets I’d treat everything after the / as the denominator. 
 
18/5

Purely because it’s an algebraic question/formula. 
 

However, in real life you wouldn’t type an equation like that without brackets. And if you wrote it down the line would be horizontal with a clear numerator and denominator. 
 

The non algebraic question would indeed  be written.

2x9/3x2-1 and give 11.

The person who set the question knows that. It’s that ambiguous internet question. 

How do you write that down for someone else to follow or review though?

Too many brackets can make code hard to read. That's just as dangerous as errors due to incorrect operator precedence.

I'd never heard of BIDMAS or BODMAS until this thread. We just learned the conventional operator precedence in maths, then in various programming languages.

Operator precedence convention is no different to all other mathematical notation that you have to learn, if you're you going to play that game.

I'm finding this thread fascinating - I don't teach maths but it's really interesting to see how different people use and interpret learning tools like mnemonics and to see demonstrations of how different people think about things.

Regarding mnemonics and similar tools in general, I remember learning them but I never found them particularly useful. For me, learning "red orange yellow green blue indigo violet" was no harder than learning "Richard of York gave battle in vain", and remembering how a rainbow looks was easier than remembering either - guess that makes me a visual learner! I do remember classmates whose entire revision strategy seemed to be committing those things to memory though, and I don't recall any of them doing significantly worse than me. 

For BODMAS in particular, I can see why it would be confusing to literal minded individuals. I don't really know what the answer is though. I don't particularly remember learning the mnemonic (as above, I never found tools like this particularly helpful anyway), although I do remember using a coloured pen to add brackets to things like 6 + 3 x 2 in tests to make the order clearer to myself. When writing code, I use brackets liberally to ensure the software is interpreting whatever I am giving it the same way that I am.

This thread is the first time I've ever heard of BODMAS. The school I went to didn't believe in mnemonics, though we were taught OHSAHCOAT. They can be useful though. When I started Chemistry, the very first thing we were asked to learn was the first 20 elements in the Periodic Table - we would be tested at the beginning of the next class. (We were forever being tested in everything!) Several minutes before the next class, someone reminded me of the test. I had clean forgotten about it. So in a panic I came up with this in the nick of time, which is so silly that I have never forgotten it:

Had He Lived Before Batterby Could Nick Our Form, Nobody Showed Much Attention. So Please Show Considerable Attention, Poor Children!

[Batterby was the name of a master at the school.]

I remember the master handing back my quiz sheet: "Well done, Stainforth, 20/20!"

If something is complicated enough that its hard to follow brackets then its liable to be a pain to follow anyway.  Any decent IDE will make it easy to apply temporary formatting to make it easier to follow. Plus, if necessary you can take it one step further and split out into variables if it really gets messy. Brackets makes that nice and easy since its just plug and play.

Don't ever go and watch a clip of it on youtube.  That theme tune is not an easy one to get out of your head once it's in there.

This reminded me of the only time I ever really found a mnemonic useful, which was learning the "reactivity series" - a subset of elements to be learnt in order of how reactive they are.  I guess you could put any elements in, but the version we had to learn had Potassium, Sodium, Lithium, Calcium, Magnesium, Aluminium, Zinc, Iron, Tin, Lead, Copper, Silver and Gold.  At that point in my education, we had learnt a lot about group 1 and 2 elements, so I was trying to apply the logic that worked there (group 1 bottom to top, group 2 bottom to top) to everything else.  Of course, that doesn't hold true as you move across the table, and as I learnt more I understood why, but at that point it was something I needed to know and I didn't have the full knowledge to figure it out logically.

My friend Anna and I made a mnemonic, which I don't fully remember, but it was something about the effect of smoking lots of pot on zebras. As I learnt more, my understanding of the actual chemistry developed and I ceased relying on the mnemonic - consequently I can now remember the order but not the mnemonic.

Agreed - and that is where I break the code into several lines with some appropriately named intermediate variables.   It's far more readable that way.  Recently I have switched to a different convoluted approach for DEQs with a great many terms and operations - I build the equations in symbolic algebra, display them typeset with LaTex for review, and have them automatically converted into executable code.  This way I have a single source for data entry, documentation, publication and code execution.  Python/Jupyter/sympy.

Jolly good!  I get the utility of an mnemonic as a teaching aid fall back that's taught to people who struggle to just learn it, but I don't get the case for putting it front and centre of the teaching.

Just to add my 5 cents to this discussion, having gone through the Greek school system in the 90's and early 00's, I genuinely do not remember ever been taught any mnemonics. Whether this means that we had none or they were really successful at replacing them with actual algebra, I cannot remember, however I do remember being taught that  2 - 1 = 2 + (-1) and 2 ÷ 4= 2 * (1 / 4). From that point onwards I am pretty sure we did not ever use the division symbol for anything. We were also not allowed to use a calculator in any physics exams, the actual answer was algebraically deriving the formula to plug the numbers into, the numerical result was not worth enough points to bother with unless you had loads of time left.

Robert did mention using the division sign to avoid doing double decker fractions but I do have to ask, why would you avoid them? It is the unambiguous way of dividing fractions that makes sense.

Would mnemonics that were all Greek to you have been any use ?

😂

I had an electrical machines lecturer who was a Geordie. So when he spoke about rotor copper loss, it sounded like the name of some Greek bloke, Rotokopolos. So, eventually, one wag got in early (I think a fellow Geordie), and wrote on the blackboard 'who the hell is this bloke Rotokopolos?'

This prompted a chuckle from the lecturer, and a long anecdote about visiting Greece, and reading all the signs as formulae...

I would hope that if the need to write it down arose there would be some form of explanation beyond a string of numbers and symbols.

This is a dangerous misunderstanding, presumably shared and perpetuated by many of the humanities graduates that dominate education and feel qualified to spout off about MINT subjects.

The end result is that too often schools teach students how to calculate, rather than how to grasp the underlying abstract mathematical concepts.

We have to deal with the fallout at uni level.

CB

I normally only remember mnemonics for their own sake, usually because they are stupid.

SOHCAHTOA: Some Old Hippie Caught Another Hippie Tripping On Acid

Resistor colour codes: Bad Boys Rob* Our Young Girls But Violet Gives Willingly...

* (other versions available)

Concepts are so much more useful!

CB

The only ones I've ever remembered are the ones I have to teach.

The most stupid one is that left hand/right hand/motor/dynamo thing. Unless you use the correct hand, you get it wrong and there is no way of remembering which hand to use! I only got it sorted once I learnmt it as a vector product.

Absolutely

You can't tell your left from your right...?

Yes, but I don't think we are there to give them an easy time. We ought to be challenging them to think and to grasp concepts. Obviously this is an ideal and we have to be pragmatic, but I think I would lose the will to go on if I ever gave up on understanding being the priority.

I'm not sure what you mean. You cannot derive the definitions from first principles! I do try to teach why trigonometry works at all (similar triangles - I get them to draw several triangles of different sizes but the same angles and to discover that the ratios are independent of the size and we then discuss why this is so). In the end the definitions need to be remembered by all abilities and Sohcahtoa seems to work fine.

Yes, far too much. I find it depressing. We are simply building a tottering edifice withoput the glue of understanding which inevitably eventually crumbles to dust. The experience of far too many pupils of maths is to be stretcherd to breaking point.

Of course I can. I just can't remeber which one to use.

Eh? You said you didn't learn which was the first one, yet you knew to start in the upper right hand side, so you did learn it!

I haven't (quite) reached the last stage but similarly in 30 years of using and teaching maths in engineering, I don't ever recall an error from incorrect interpretation of a written equation, or any confusion in a practical case.   As others have pointed out, in programming (and excel) it can be a problem but in these cases the answer is brackets and suitable variables, not "BODMAS".

Yes, that is pretty much what I do. I ask the whole class to write down the answer to 2+3x4 and we then discover that they are split between 20 and 14, so we discuss the need for a convention to aavoid chaos.

Use a handy (geddit...?) mnemonic:

D is for Dad and Dynamos

M is for Mum and Motors

Dad is always right...

As if I'd remember that lot. what ius the p;oint of a mnemonic that is harder to remember than the thing it is meant to remind you about! I think I'll stick with the vectpr p[roduct.

I've rationalised the resistor codes over the years into an idea of increasing brightness - black, brown red orange yellow fairly obvious and I suppose it its easy to think of blue as brighter in the energy spectrum than green, then purple/violet as in beyond the blue and white is the brightest colour (to us) with grey being slightly less bright.  Been using that picture since the 1970s and the only time I get it wrong is when you find tiny things with orange looking like red and vice versa.

the only weakness is enforcing blue as higher brightness than green.  It;s a bit idiosyncratic

it amused one of the teachers when I showed him how I could test a transistor and tell whether it was PNP or NPN using my tongue, and even give an indication of whether germanium or silicon and gain using a wet finger as well as the tongue to apply base bias

I remember that the left hand is for the load.

20 years since I used that, and it came back as soon as I put my hand out in the position.  

Lost me there........

Like many mnemonics, you don't have to remember the whole thing, or even the mnemonic at all; they're often a means to establish the memory.

So you only need to remember D (dynamo) is right.

Do you use the hand mnemonic for remembering the rules?

Those rules are taught because the electromagnetic field rules are usually taught, at a practical level, long before vector products are introduced in the contemporary mathematics classes.

I'd never heard of it before, but it seems pretty obvious: A motor has a Load.

It's not a great mnemonic, because a dynamo can be a load, so it's open to confusion.

The motor is the load and the right hand is for generators I think.  Is that right?  

Had to Google this as despite having degrees in Mathematics and CompSci, I have no idea what you're talking about.

Of course, the really silly thing about this is that Fleming had the direction of current wrong, so if you go by electron flow then you need to switch hands anyway...

That's conventions for you. Pesky things...

Precisely. I can't. 

I do seem to have a blockage with 2 way arbitrary choices. I have to use trial and error every time I use a screwdriver and if I think too hard I struggle I can get taps wrong as well.

Yes, I found things got easier once I learnt about vector products.

Agreed.  Which is preposterous when EMag is all so simple and clear if approached from the outset as vector fields instead of the tooth pulling approach of deriving the field from Coulomb’s law and the whackadoodle stuff around left and right hands and so on. With Coulomb’s law  itself presented with no intuitive reason to be as it is. Unlike just popping out of the first principles of geometry, fundamental charge and superposition in a linear field theory.

With the various rules, my preference is to teach the vector product and the formula using it. Then one hand is only ever used to define the vector product, and I make the point that either handed coordinate systems can be used as it is an arbitrary convention that cancels out of equations of motion etc, but give the one commonly used.  For people who don’t take to this I also give a simple system for computing a vector product from base vectors based on rotational permutation of the first one.  Then they’re always rote memorisation of their 3 permutations.  Everyone should “get” one of those.  Again though this is introductory undergrad not school.

Ah.  I do have a useful one for screwdrivers.  

Lefty Loosy

Righty Tighty. 

Okay, so you struggle with certain learning tasks. Do you think all your pupils will have exactly the same mental block? Or maybe a different mental block? One that might be overcome with a different approach to the one that seems 'correct' to you?

To return to the trig functions issue; I teach navigation to DofE participants. Bearings are interesting, compared with mathematical angles, since bearings are measured in the opposite direction, from a different origin. Conventions, again, that have to be learnt.

A mnemonic commonly used for grid references is 'along the corridor and up the stairs'. I don't like this, because theres no reason why 'up the stairs and along the corridor' should not be equally plausible.

Then there's the abomination 'grid into mag, add; mag into grid, get rid', which is gibberish to start with, not universally applicable, and not even 'correct' in Britain any more... The fundamentals of knowing the relationship between magN and gridN are universally applicable, unambiguous, and just as easy to remember. Theres even a handy mnemonic figure on the map...

You might as well argue that tensor algebra makes so many problems so much easier to explain.

Trouble is, tensor algebra is somewhat advanced stuff; we didn't cover it in undergrad E&EE, even though it does, I understand, make EM theory much more elegant.

We teach bearings in Maths, not that you'd ever know it.  

It's one of those topics, not especially difficult, but it just doesn't seem to stick.  Maybe we just don't give it enough time and practice to get into long term memory. 

But you don't turn a screwdriver either left or right, you turn it clockwise or anticlockwise, so that simply doesn't make sense!

If you grip it like any normal person your fingers pretty much move left or right.  Don't try it with  bike pedals however, it will only work on one of them...

Of course. But I am absolutely convinced that, whenever possible it should be based on understanding and obviously there are differing approaches to explanations. However, in the case of arbitrary conventions, there simply is nothing to understand, so we have to resort to  silly mnemonics and so on. I would encourage pupils to use whatever works for them. If they want to use CAST or SATC that is fine, but I will certainly make a point of explaining why it is logically bollocks (which amuses them and might even promote understanding).

Edit: Of course ASTC is not arbitrary, but sometimes one had to be pragmatic.

So how do I remember whether it's thumb or fingers. It is just ridiculous to think in terms of left or right. Why not come up with a mneumonic for clockwise or anticlockwise?

I take your point, but my rant was aimed at foundation undergrad courses and textbooks that tend to start with Coulomb and end with fields; unlike your tensor point, everything is covered at the relevant level, it's just taught arse backwards because Coulomb is what happens to be taught at 6th form.

It's the bit of your hand you see at when using a screwdriver or similar!

In a theoretical sense yes, in a practical sense it works well.  Anyone actually using a screwdriver understands it instantly.

Not if you are kneeling on the floor putting a screw into a skirting board.

So it works for somebody who uses a screwdriver regularly, but they presumably do it on autopilot anyway. It is useless for very occasional users like myself who can't do it on autopilot.

I think you are an exception.  It worked fine for my mother, for example. That said, if you can think of a catchy version with clockwise and anticlockwise, I am sure it would catch on.

Sorry, I can see why that looks contradictory.

I use the mnemonic to remember to write SATC in the four quadrants, in normal left to right fashion, rather than cyclically. This mnemonic helps me land the correct letters in the correct quadrants, with the A in the upper right. A stands for "all" and it isn't the initial of a trig function, so start on that one. Being in top right means anticlockwise is the "shortest route" to my vector from the horizontal, this reminds me to continue in the anticlockwise direction to get the other solution from the other relevant quadrant.

Opposed to: remember CAST labels anticlockwise from bottom right. (Or ASTC from top right.) This is how we were taught and feels alien because I'm not used to writing in circles starting away from the upper left of my page. I write in rows from the top left.

I was replying to:

I took that to mean I need to know to "start ASTC labelling top right", which was exactly what I struggled to learn. If you meant that you still need to know to start finding solutions with A, then I agree.

My point was I didn't have to know the location of the first quadrant or the direction to go in, I remember a mnemonic to label them up, and "know" to start on A, this reminds me this is the first quadrant and a little extra rule reminds me to go anticlockwise to get the other solutions. I dont know to start upper right and go anticlockwise, I know a silly phrase and to start on the odd one out, which I was suggesting is a slightly different thing to remember, and was easier for me to learn.

If it's  your thumb you see, it would still be moving L-R to tighten so it would still work.  Try using it and see you have a problem.  Or like most people just turn it the correct way!

Generally anti clockwise is initially to the left and clockwise is initially to the right.  

Think of it like bearings, you have a North to the top of the page and you turn clockwise, so you go right from the North line. 

I've just tried it with variously positioned hypothetical screws. Whether my thumb or fingers initially move right or left is entirely dependent on the position. It's bollocks!

What is? 

Or left from the south line, or down from the x-axis. Working with clockwise and anticlockwise is the only sane way to do it.

I really think you are looking at it much to theoretically.  It's a practical rhyme to act on instantly when you have a stuck screw, not something to ponder.  If it didn't work, people wouldn't keep using it.

It this is typical of your thought processes when using a screwdriver, you really aren't like most people!!

Pretend you are the screwdriver.  If you stand up and you turn clockwise you turn to your right.  

If you turn anti clockwise you turn to your left.  

Robert Durran after a frustrating evening trying to put up some shelves: "F*ck it, pass me the hammer!" 😠🔨🚑

It is quite simple. A screwdriver turns either clockwise or anticlockwise (unless you are the screw). The very fact that you lot are having to invoke all this external dodgy nonsense about fingers, bearings, people turning to make any sense of this left/right bollocks simply confirms that it is a ridiculous way of thinking about it.

Yes, I seem to be uniquely rational and sane.

I was taught the resistor colour code for the first time by a 60 year old chap from my local amateur radio club. He used a horifically racist (and utterly inappropriate) mnemonic to remember it and I could never get anything less horrendous to stick in my mind. I remember standing in front of my boss trying to find the right resistor from a group of 5 without muttering it under my breath, much safer to get a multimeter on the job!

If someone tells you to look left, do you turn your head or body 90 degree turn anti-clockwise, or does your very rational and sane self turn 270 degrees clockwise? 

You are a genius.  I’ve spent this evening wracking my brains as to why I associate counter clockwise with left.  Now I know!

But what if I have been using digital clocks all my life and dont really do that clockwise anti clockwise thingy?

You're screwed.

You seem to be particularly dogmatic...

"Remind me, which way do I turn this bloody hammer to get the nail in?"

But if I want to fart to my left, I turn clockwise. 

That depends if you look down from above your head, or up from under your feet...

I know which way to turn a screwdriver to tighten and loosen without having to use a rhyme.  It's just logical and instinctive.  
What I think people are amazed at, is that you don't! 
As for 'Righty Righty Tighty Tighty, Lefty Lefty Loosey Loosey'.  It makes perfect sense to most people. Perhaps you're the one with the problem?

 

Far out in the uncharted backwaters of the unfashionable end of the western spiral arm of the Galaxy lies a small unregarded yellow sun. Orbiting this at a distance of roughly ninety-two million miles is an utterly insignificant little blue green whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea.

I would make the wild guess that people do not define their forward vector by the direction of thrust produced by the digestive system.

Edit: I have of course got that vector wrong by 180 degrees, which would mean that most people *do* in fact define their forward vector by the direction of thrust produced by the digestive system, rather than the direction of... the ejecta.

That's from Rocky Horror, right...?

What if you're someone who struggles with remembering which way clocks go round?  Maybe you're most often exposed to time in a digital format, or grew up in a house with a novelty clock with all the numbers on anticlockwise.

The method that works for the individual is the only sane way for that individual to do it.

You've used the words silly and ridiculous in your replies. I hope you're not actually in the mindset that your students are worthy of ridicule, or tried to make them feel that way when pointing out their methods of learning are "logically bollocks".

Of course you want to share with them the deeper understanding of how it all really works, but until the topic is familiar, some need a stepping stone. I know I did.

The mnemonics are only silly when they dont work for the individual, per OP, per my experience of BODMAS before grouping DM,AS.  Per seemingly, your experience with every mnemonic because you're able to grasp all incoming abstract concepts immediately.

Ah, you beat me to it!

It's not logical and it's not instinctive, though it may well feel instinctive if you use a screwdriver frequently. It's an entirely arbitrary convention. 

Obviously I have a practical problem*, but I prefer that to muddled thinking.

It is only a mild problem with a screwdriver because trial and error only takes a second or two. It's more of a problem when turning off a flaring petrol cooker in a hurry.

It's logical because you assume that clockwise is positive (tighten) and and anti-clockwise is negative (loosen).

You would have to come up with another way of remembering it. Maybe roundabouts or the the little twirly thing when a computer is thinking.

Absolutely (see above).

I have no problem with silly mnemonics if they work for people.

I am quite happy to point out that SATC or "righty tighty" are logical bollocks (in fact I feel obliged to do so). It is nogthing to do with ridicule. As I said earlier, a good rant as to why they are bollocks might be entertaining as well as illuminating. But, in the end, if they work for them I have no problem with their use. 

As I have already said, I have no problem with pragmatism when necessary.

If you were taught it without the grouping being made clear, then it was poor teaching.

Good grief. You're really clutching at straws aren't you. Are you seriously suggesting that is the reason for the convention? It's just another contrived mnemonic.

We all have minor practical problems like that. For some reason however, you are trying very hard to rationalise that problem by claiming that a very reasonable mnemonic that helps with it to be muddled thinking. It is not. Turning left is turning anti-clockwise for the very obvious reason that this is the direction a person would turn themselves, it is based on actual every day human experience. Yes, it does break down if you are using a screwdriver in any orientation not aligned with your face, but that doesn't matter because just remembering left is enough to recall anti-clockwise. It is not a mnemonic I use, mainly because I have been able to use a screwdriver for longer than I've been able to say left, right, clockwise or anti-clockwise but it is nitpicking to claim it makes no sense. You are trying to find reasons for it not to make sense.

PS: You really should not rely on trial and error when figuring out which way a fastener is loosened. If they are stuck or just thread locked both rotation directions will feel the same but one of them can cause damage. 

Ok. fair enough. You have come up with a reason why left might be associated with anticlockwise. But the fact is that everyone else was coming up with complete non-reasons.

Edit: I have actually had this discussion about "righty tighty" on several other occasions (the first occasion provided an entertaining evening at a base camp in Greenland after I had to ask which way to turn the knob to turn off a cooker). I have never had anyone else side with me (which I genuinely find extraordinary), but you are the first person ever to come up with a reasonable explanation. Are all the others subconsciously associating left with anticlockwise for the same reason without being able to articulate it?

Some of the recent posts way down here in the thread made me realise that perhaps UKC should play to its stengths here.  After all, you didn't post this on a teaching or maths forum...

I return to the forums to waste hours of my life partly for the climbing content but also because I can be sure of a laugh.  Some of these guys have just made me chuckle.  The HHGTTG reference, the Rocky Horror reference, Robert's "twirly computer thinking thing", fart jokes.

Now I greatly enjoyed making dnf acronyms out of rude words in a thread last week.  So I offer the UKC BODMAS caveat that I just dreamt up:

All Sums (and) Subtractions Happen (in) Order (from the) Left End
I'm not totally happy with its clumsiness, but being introduced a phrase based on ASSHOLE isn't something I'd forget in a hurry.

Any other novel suggestions out there?  We've done the sensible teaching and maths aspects of the subject to death now!

Obviously one “does the bottom second” in the quotient rule. 

I’m not saying I remember it like that, or with any posterior related imagery, but one could. Couldn’t one?

Clockwise increasing and anti-clockwise decreasing isn’t convention. It’s observed by the progression of the Earth round the sun. 

How is that either increasing or decreasing?

What? In what way is that meaningful for astronomy or screwdrivers?

I never heard 'righty tighty etc' until a few years ago, it was always clockwise and anti clockwise (although right and left hand thread was used ..).

Does it make more sense to you with a spanner/ratchet rather than a screwdriver? A right handed person would pull to the right to tighten and push to the left to loosen?

No. 

So what does work for you?

The only reason I've resorted to clutching at straws is to try and supply you with a description that will satisfy your bizarely over logical brain.
As for it being the reason for the convention? Haven't got a clue, but engineers generally do things for a reason, so making screw threads tighten when you apply positive degrees of rotation does have some logic to it.

The numbers on my clock increase as they go clockwise. Yours may be different I suppose. 

Ah! RPN - reminds me of my trusted old HP 15C. It must be in a drawer somewhere. Not used it for years as phones a small powerhouses these days. But wouldn't it be nice if their calculators could be set up to do RPN?

Again, that depends on the point of view; looking 'down from the head', or 'up from the feet'... Just like all this starships all being the same way up...

Our 'clockwise' direction is based on the movement of the shadow of a sundial.

Screw threads are another convention that has to be learned. Except in some circumstances where the thread sense is determined for operational reasons; bike pedal and bottom brackets, for instance.

Just checked mine. The sneaky things just fight for the same space.

I'm really very happy to use mnemonic code assembly language rather than just the machine code.  Though A92A2075FE4c0004 works for me too

Clockwise is negative rotation, as far as mathematicians are concerned... Another arbitrary convention that has to be learned.

They do.  It was the bit about the sun and earth I was querying.  I don't see how that is positive or negative, or even clockwise or anti-clockwise

You can get emulator apps for the HP calculators.

Sundials. That’s where we get clockwise from. 

11, 12, 1, 2 isn't an increasing sequence 😉

What about sundials in the southern hemisphere?

(Just taking over from Robert while he has a brew)

That is indeed weird if you look at it like that. Polar co-ordinates. Although it’s essentially a function of angles getting bigger rather than any clockwise/anti-clockwise convention. The base of a triangle is horizontal and the apex is higher than the base. So the included angle is +ve ‘anti-clockwise’. So I wouldn’t say that’s convention, it’s the real world application. 

Very good. Blame the Egyptians for their superstitions. 

They go anti-clockwise. Clocks would go the other way if the aborigines had invented the clock. But they didn’t. It’s not convention other than they kept everything the same based on sundials. So you could argue tenuously I suppose, but that’s really stretching it. 
 

Checking your watch against the local sundial would be a bit of an exercise. 

You could just as easily draw the triangle the other way around. It's an arbitrary convention.

Why do we write L-R (or R-L, it T-B)? Why does the Cartesian coordinate system have positive to right and top?

There is a chirality to many molecules in nature, and it’s always the same.  The choice doesn’t appear to be arbitrary.

https://www.sciencedirect.com/science/article/abs/pii/016561478690235X

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2857173/

Why would you? Mostly they were using triangles to look up and triangulate heights and stars. Hence triangles have a ‘base’ rather than a ‘top’. 

I wonder how many galaxies are clockwise and how many are anti clockwise. 

Why wouldn't you? There's no 'right' way to draw a triangle. It's an arbitrary choice to have the perpendicular on the right.

Fair point but they are in fact connected with the rotation of the earth about its axis, not about the sun.

This thread isn't doing my Top 40 any good at all...

They are all both. Again, depends on the point of view. Looked at from the back, a clock hand turns anticlockwise.

The base is on the bottom. You look up so you are looking up to the apex. The perpendicular drops down to the base. Doesn’t matter what way you face. The angle increases as the apex gets higher and the base length increases as the apex gets further away from you. 
 

The only arbitrary thing about the polar co-ordinates is the X axis runs left to right. Because we read left to right and numbers increase left to right. 

But is it a right or left hand thread, that's the important thing.

Draw the triangle with the perpendicular on the left, and the angle in question on the right. Now, angles increase clockwise. Arbitrary convention.

Arabs don't.  Nor do the Chinese

They decrease l-r 

Only works if you’re looking down off a cliff. Very rare when you’re in a boat. 

More importantly how is it stored? Is it big endian or little?

How about when you reach the edge of the world or are you one of those round world nutters?

Works perfectly well if you're plotting a star to your left...

Chiral pairs are also common. It's not always the same. Spearmint and caraway have smells as a result of the different enantiomers of the same molecule.

The stars are only on the dome above. 

That's something interesting I've learnt today.  Thanks 

Who else is using on their iphone? I do, RPN out of pure nostalgia!

CB

I live and learn!