Dead easy — most of time the question has simple numbers that give the answer at once, and in the rest you only have to estimate to more than the first significant figure in one case.

Arnald

Oh dear.

Anyone point to an equivalent test 40 years ago?

bloke in spain

The Sage is doing something seems lost on many people today. The rough approximation. Given that, you can asses whether an answer is realistic. I’d reckon I do the same with most every calculation I make.
Yet how often do you see figures quoted in the papers at complete variance from the underlying information? Often complete orders of magnitude. It’s not just an inability to do simple arithmetic. It has wider implications. It’s failing to reality check assumptions. Why so many people seem to hold beliefs don’t make any sense.

The Other Bloke in Italy

Arnald, as you say: Oh dear.

I recall doing this sort of thing (I thing it was called “Problems”) last couple of years in primary school.

So, about 1960. I was considered slow in those days.

Alastair Harris

If you use a calculator you are missing the point.

Chester Draws

Often a substantial barrier to these sorts of things is understanding context. Most of us have travelled in foreign parts.

Try explaining exchange rates to a kid who has never left the country as I do. It’s quite awkward — they don’ t understand the whole concept of exchanging money.

These tests are easier for older people as a result.

The 10th question using reverse percentages is a real test of mathematical comprehension. You can waste hours trying to teach kids how to do this, yet a good kid will understand immediately. It is not a test of education though. You either get them or you don’t.

Chester Draws

You never did reverse percentages in primary school. You are kidding yourself.

Pat

Let’s have a stab at this.
Universities sell an increase in human capital to people. The universities themselves get to certify that said human capital has been delivered.
Is there the slightest possibility that they may either sell courses that fail to raise human capital by the advertised amount, or offer places to people who cannot absorb the human capital on offer?
Given that they certify delivery.

Tim Worstall

This is something that I (as does Brad Delong) rail about repeatedly. Just general ideas about economic numbers. GDP is about £1.5 trillion. Median wages are mid-20ks a year. Some 65 million people in the country. Just orders of magnitude stuff. And yet we really do read econ articles that just don’t seem to have those basics at their fingertips.

Arnald

bis

I agree, one of mine is starting GCSE this year, I’ve helped on a couple of things – expanding double brackets and the like (I seem to remember them from my o level), but I haven’t seen the estimated problems. It is an important skill.

Also I noticed there were easy fraction problems, that can’t have been in o level, surely.

Kids do seem to struggle with the applied aspect, though – Peter and Paul and their amounts of apples and oranges stuff. They’re fine when you get them to x and y the variables so it just becomes reading comprehension.

I agree with CD, reverse percentages are a step up, and they’re also applied regularly in everyday life. If you like to work out grocery deals.

Interested

@Chester

My kids definitely did ‘reverse percentages’ at primary school (4/5 years ago) – I can place it precisely because we moved house when my youngest was eleven and I remember testing them on the whole lot in our previous house.

These are trivially easy questions. It is a little concerning, though, as I say, my kids did this sort of thing ages ago. I suppose, as ever, it’s about the schools you attend?

Interested

re O levels/GCSEs

There’s twelve years between my younger sister and my brother; when he was doing his A level maths (having got an A or A*, if they existed then, at GCSE) he used her O level Letts Revise study guide and found it very tough. Same with biology.

He’s thirty one now, so (this suggests) grade inflation was a thing then.

(They both attended the same highly academic private school, results consistently in the top fifty, and in each case had exactly the same teachers.)

Dongguan John

Interested. When I did A level maths in 95/96 the teachers gave us old O level papers to practice on.

Ed Snack

I have to say that in the “old days” I reckon I could have solved the lot in a minute or so. That’s back in the days when desk calculators were a rationed resource and accounting consolidations were done on 14 column paper. Now everything is in Excel, some skills atrophy if not used a lot, and although I could answer the lot fairly quickly, I don’t think I’m as quick as I was.

It used to be sort of fun in an odd way to take one of those open ended sort of tests they used to give for some job interviews. The sort where they say “answer as many as you can in the time, you won’t finish it so don’t worry”. So you could make a point of nonchalantly handing it in with a couple of minutes to spare saying “I finished it OK….”

Surreptitious Evil

The only one of those that is non-trivial is no 9 (and it didn’t help on the iPhone that I had to guess the end of the question!)

But you don’t need to work it out because only one of the four options is even approximately right.

raj

My son is in the first year of secondary school (i.e. 5 years from doing GCSE’s ) and I would be very surprised if he did not get at least 9/10.
Whilst he enjoys maths I would hope the majority of his classmates could also get well over 50%.
Therefore the idea that University students could not answer almost all the questions horrifies me.

Jason Lynch

BiS,

Those of us of a certain age, who weren’t allowed calculators in a Maths O-level but *were* allowed slide rules – which put a major premium on understanding what order of magnitude your answer should be (as well as a healthy scepticism about too many significant figures of accuracy – I’ve seen fellow students painstakingly work out that an answer is “14.342593 plus or minus 21.843875%” because if the calculator shows eight digits, it must be that precise) had to learn the “looks about right” estimate. I flashed up a calculator for the restaurant question, for example, because there were two credible options for the answer – but I could get rid of half the options at a glance.

More generally, when recruiting candidates I had to weigh A-level grades by a rule of thumb: an older candidate with BBC from the 1980s was probably as good as someone with ABBA from the late 1990s.

There was real concern in 1987-88 among those teaching subjects like A-level chemistry and physics that the new GCSEs wouldn’t adequately prepare students, and IIRC they were proved right – stuff we did at O-level is now considered A-level material, subjects we covered at A-level are now moved up into a BSc.

It runs right through the system. Field theory that I did as a fresher is now second year material, but only in an “option”. How the hell you can be a physicist without field theory beats me.

jgh

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Ben S

Didn’t like the restaurant question because it said ‘exactly’, and you can’t divide 3p by 4 exactly.

monoi

My boy in y6 primary school already does all this and more.

10/10 and I worked each out before looking at the multiple choices. For many questions it would clearly have been easier to look at the choices first as knowing the approximate answer would be enough to choose the only plausible option.

The restaurant question is unrealistic: if you want to share the bill in such circumstances split the amount to a convenient round number and round up the contribution to make up the tip. ie 4×22 covers the bill, so everybody pays 23 or 24 if you like the service.

Chris Miller

Maths tests down the years

1960s: A forester sells a load of timber for £100. His cost of production is 4/5 of the price. What is his profit?

1970s: A forester sells a load of timber for £100. His cost of production is 4/5 of the price, or £80. What is his profit?

1980s: A forester sells a load of timber for £100. His cost of production is £80. Did he make a profit? (Answer Yes or No )

1990s: A forester sells a load of timber for £100. His cost of production is £80 and his profit is £20. Underline the number 20.

2000s: A forester cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands. He does this so he can make a profit of £20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the birds and squirrels feel as the logger cut down their homes? (There are no wrong answers, and if you feel like crying, it’s OK).

PF

CM

Can’t wait for the 2010’s ‘maths’ test… Perhaps by then the squirrels will have re-identified as gerbils (or hamsters), and the logger, now that she is considerate and caring, will adopt them as pets.

Calculators are allowed? That’s the shocking bit. You shouldn’t even need a pencil for these.

bloke in spain

I can recall the absence of calculators. Apart from those mechanical wonders full of cogs you turned like a coffee grinder. I started my working life on the stock exchange calculating in shillings, pennies & farthings, to exact figures, in my head. Since then, everything I’ve done has required maths. Including navigating an aircraft. Still rarely use a calculator, although I like slide rules.
I got so low a result in my mock O-Level, small single figures, they wouldn’t let me sit the exam. Bothered, I wasn’t.

The Sage is doing something seems lost on many people today. The rough approximation. Given that, you can asses whether an answer is realistic. I’d reckon I do the same with most every calculation I make.

Absolutely. This drives me nuts at work, the French just cannot do approximation. Also, very few understand the maths behind that last question regarding the £108 price tag which has already been reduced by 10%. I have supposed engineers who mastered in maths telling me that if Y is 80% of a total I need to add (Y x 20%) to reach the total.

When I did A level maths in 95/96 the teachers gave us old O level papers to practice on.

You, sir, would have been sitting those exam much the same time as me.

MyBurningEars

“You never did reverse percentages in primary school. You are kidding yourself.”

Possibly not. It is taught at primary school though perhaps not at all primary schools and not to all kids. But it is a classic 11+ question so for regions with the 11+ it will certainly still be taught; moreover it is in year 6 SATS papers.

Grikath

2010s: The teacher does not understand the question, and has dispensation for Dyslexia.
He conveys this to the class by interpretive dance, and proceeds with todays planned lessons regarding the relative merits of using recycled bioplastic in this weeks scrapbooking assignment about the feelings of logging-related PTSS-affected squirrels.
Johnny, however, gets a Polite Warning and a trip to the Headmaster for a Mild Admonition for looking at the paper on the teachers’ desk and attempting to answer the Questions.
A parent-teacher meeting with a Mediator is held to adress Johnny’s inability to Assimilate in the Group, and on careful consideration it is decided that Johnny should have remedial therapy to alleviate his tendency to asocial behaviour.

I could go on, but you get the picture..

Tractor Gent

This is the sort of stuff you would test an entrant to a trade course on at the local college to see if they needed a remedial course first. Of course the local colleges are all ‘Universities’ now but presumably the entrance requirement is no higher:(

My O level maths exam in 1965 included calculus – both differentiation & integration. And both were necessary for stuff we did at A level.

This isn’t good preparation for a STEM degree – it needs several more levels of difficulty, especially for anything that is based on physics, i.e. just about everything. I would also ask rather more than this for a humanities degree too, given the dire rubbish we see nowadays from journalists, pols & ‘activists’.

Interested

Raj – I hadn’t read the piece, just went straight to the test. I didn’t realise university students were included too. Yikes.

One thing I would say – I used to be a big believer in people being able to do a lot of this stuff in their heads, calculators ought to be verboten in exams etc, but a friend of mine changed my mind by simply pointing out that we will pretty much always have calculators now. (Very obvious, it just hadn’t really occurred to me in the way he said it.) They’re ubiquitous and will stay that way, I’d have thought, so maybe we don’t need to worry about doing stuff in our heads. Though I do think there’s a satisfacttion, and possibly some fringe benefits in logic etc, to be picked up that way.

(I know there are jobs which require mental arithmetic and head maths and always will, I’m on about the generality.)

Chris Miller

@Interested – I agree that there’s little benefit in being able to multiply 4-digit numbers in your head to 7 or 8 digit accuracy, But it’s important (if you’re relying on a calculator) to at least have a feel for the order of magnitude of your expect answer and perhaps the leading digit. That way you may spot that you’ve accidentally transposed two digits or pressed – when you meant +.

Don’t get me started on Excel. I remember reading a survey by one of the large accountancy firms that claimed 75% of the spreadsheets they audited contained errors, in my experience that’s a low estimate. But if it comes out of a computer it must be gold, innit.

@BiS – I started my actuarial life working with a Monroe ‘coffee grinder’. Supervisors got electric (not electronic!) models, but the real pros could ‘twiddle’ faster. Later I graduated to a programmable Epic 3000: http://www.vintagecalculators.com/html/monroe.html

Bloke not in Cymru

I’ve been teaching my daughter about approximation, like many I did o level with no calculator and approximating was part of the process for us, also spotting known whole number solutions (Pythagoras solutions for whole numbers 3,4,5 etc.)
As for not needing mental maths as calculators are ubiquitous then that leads to the problem highlighted of writing down the answer just because that’s what the calculator said without understanding it.
Multiple choice made that test easy, though I worked most of the answers out before even looking.

I won a Hewlett Packard LED calculator (one of the non-programmable ones) in a maths competition for primary school kids in the late 1970s.

It wasn’t allowed anywhere near school once they’d taken the photograph of it.

A few years later, I was using graphing calculators in exams.

Jim

I was definitely taught some quadratic equations at primary school, simple ones, learning how to expand and contract bracketed equations, that sort of stuff. That would have been early 80s, State primary school.

john77

@ Interested
What happens when your calculator has a flat battery and gives the wrong answer? I got talked into buying a calculator in my late 20s and a year later nearly got caught out – but the answer just looked wrong so I stopped and slowly (being badly out-of-practice) did the sum in my head.

Interested

Chris – Agree completely re estimating and orders of magnitude etc. When I did the above test I did all but the final two by looking at the answers and working out which was most likely.

John – if you work in a role which requires a calc you’ll have batteries and/or back ups. Or will be able to get them.

As I said there are roles where you will always need raw mental skills eg artillerymen need to be able to calculate in their heads, pilots need to retain ability to navigate in a non GPS world, but for most people it’s just not needed any more, or not to the extent it once was.

I used to take your view but no longer do. More important IMO for kids to get to know how to use calc than how to do it longhand.

Chester Draws

I know some standards have slipped in recent years, particularly round mental arithmetic, but some of the commentators here are alleging that students now are four years behind what they used to be!

For example, doing quadratic factorising at primary school, or doing reverse percentages, dividing by 0.9, at primary school. Without calculators mind.

I’m sorry, but that is not credible. I’d like to see some evidence of these amazing allegations.

People are not stupider (actually, thanks to better food and health IQs are rising) and teaching hasn’t changed that much — there are still schools that teach effectively exactly like they did 50 years ago, mostly private.

If what some of you say were true then when Britain was tested on the international PISA testing you’d not just be behind the likes of Shanghai, but you’d be on a separate page.

I think it much more likely, indeed certain, that many of you have rosy tinted visions of what was done in the past. (And having some smart kids briefly examine some extension stuff to keep them busy is not the same as it being taught at that level.)

I know for a fact, that New Zealand’s secondary school kids do not do a program vastly different from what they did 40 years ago. Actual fact, because I have the exam papers from back then.

(And don’t scoff that us colonials are behind you, because our PISA results are better. And we have more immigrants than you, so that’s no excuse either.)

Chester Draws

What happens when your calculator has a flat battery and gives the wrong answer?

What you are ignoring is opportunity cost.

If we spend years drilling high school kids on long multiplication and long division then they are not learning something else. All for a skill that they don’t need. Why not teach them Latin while you’re at it?

Which is different from not teaching them to approximate to check their answer. I teach it consistently, and they ignore me equally consistently.

jgh

While saving up for a calculator in the mid-80s I worked out how to do trig with pencil and paper – and no tables! I just remembered constants for SIN 0deg, 30deg, 60deg, 90deg, then did straight-line interpolation. eg, SIN 50 is 2/3 of the way between SIN 30 and SIN 60. Good enough for one decimal point of accuracy.

Bloke in Costa Rica

CD: I did matrices, basic trig, factoring binomials (using the polynomial remainder theorem), elementary Euclidean geometry proofs and base conversion at prep school in the 70’s/early 80’s.

All of those problems should be doable in your head. Fiddling with mental arithmetic keeps you sharp. I was walking into the supermarket at lunchtime dividing 745 by 11 in my head because I didn’t want to look like a twat by pulling out my phone. It’s obviously 67 8/11 and 8/11 is easy. If you want to multiply two digit numbers and you can remember your squares up to 99 then you can usually do difference of two squares plus a fix-up if the difference is odd. (10 x + 5)² is just 100 (x² + x + 1/4), and so on.

The Sage

Now here’s a poser — if you were to answer this multiple choice question randomly, what would your chance of getting the correct answer be

A 50% B 25% C 0% D 25%

Interested

BiCR

Prep school (in my experience) goes to thirteen though (in case you’re right on the school but wrong on the year?).

john77

@ Interested
Of course I could afford to buy a new battery – but it’s when I didn’t know that I needed one that the problem arose.
@ Chester Draws
You seem to be dismissing my point in one sentence and supporting it in the next. Approximation is what tells me if the calculator has failed or if (more likely) I’ve fat-fingered – the latest freebie calculator I was given I can depress four keys at once with a finger (six with my thumb).
Lingua latina docet fons linguae novae.
@ BiCR
That’s mildly impressive. I didn’t do matrices and I only did trig because the maths master wanted to keep the scholarship set interested (and quiet) after our exams while he coached the rest for CE.

Dead easy — most of time the question has simple numbers that give the answer at once, and in the rest you only have to estimate to more than the first significant figure in one case.

Oh dear.

Anyone point to an equivalent test 40 years ago?

The Sage is doing something seems lost on many people today. The rough approximation. Given that, you can asses whether an answer is realistic. I’d reckon I do the same with most every calculation I make.

Yet how often do you see figures quoted in the papers at complete variance from the underlying information? Often complete orders of magnitude. It’s not just an inability to do simple arithmetic. It has wider implications. It’s failing to reality check assumptions. Why so many people seem to hold beliefs don’t make any sense.

Arnald, as you say: Oh dear.

I recall doing this sort of thing (I thing it was called “Problems”) last couple of years in primary school.

So, about 1960. I was considered slow in those days.

If you use a calculator you are missing the point.

Often a substantial barrier to these sorts of things is understanding context. Most of us have travelled in foreign parts.

Try explaining exchange rates to a kid who has never left the country as I do. It’s quite awkward — they don’ t understand the whole concept of exchanging money.

These tests are easier for older people as a result.

The 10th question using reverse percentages is a real test of mathematical comprehension. You can waste hours trying to teach kids how to do this, yet a good kid will understand immediately. It is not a test of education though. You either get them or you don’t.

You never did reverse percentages in primary school. You are kidding yourself.

Let’s have a stab at this.

Universities sell an increase in human capital to people. The universities themselves get to certify that said human capital has been delivered.

Is there the slightest possibility that they may either sell courses that fail to raise human capital by the advertised amount, or offer places to people who cannot absorb the human capital on offer?

Given that they certify delivery.

This is something that I (as does Brad Delong) rail about repeatedly. Just general ideas about economic numbers. GDP is about £1.5 trillion. Median wages are mid-20ks a year. Some 65 million people in the country. Just orders of magnitude stuff. And yet we really do read econ articles that just don’t seem to have those basics at their fingertips.

bis

I agree, one of mine is starting GCSE this year, I’ve helped on a couple of things – expanding double brackets and the like (I seem to remember them from my o level), but I haven’t seen the estimated problems. It is an important skill.

Also I noticed there were easy fraction problems, that can’t have been in o level, surely.

Kids do seem to struggle with the applied aspect, though – Peter and Paul and their amounts of apples and oranges stuff. They’re fine when you get them to x and y the variables so it just becomes reading comprehension.

I agree with CD, reverse percentages are a step up, and they’re also applied regularly in everyday life. If you like to work out grocery deals.

@Chester

My kids definitely did ‘reverse percentages’ at primary school (4/5 years ago) – I can place it precisely because we moved house when my youngest was eleven and I remember testing them on the whole lot in our previous house.

These are trivially easy questions. It is a little concerning, though, as I say, my kids did this sort of thing ages ago. I suppose, as ever, it’s about the schools you attend?

re O levels/GCSEs

There’s twelve years between my younger sister and my brother; when he was doing his A level maths (having got an A or A*, if they existed then, at GCSE) he used her O level Letts Revise study guide and found it very tough. Same with biology.

He’s thirty one now, so (this suggests) grade inflation was a thing then.

(They both attended the same highly academic private school, results consistently in the top fifty, and in each case had exactly the same teachers.)

Interested. When I did A level maths in 95/96 the teachers gave us old O level papers to practice on.

I have to say that in the “old days” I reckon I could have solved the lot in a minute or so. That’s back in the days when desk calculators were a rationed resource and accounting consolidations were done on 14 column paper. Now everything is in Excel, some skills atrophy if not used a lot, and although I could answer the lot fairly quickly, I don’t think I’m as quick as I was.

It used to be sort of fun in an odd way to take one of those open ended sort of tests they used to give for some job interviews. The sort where they say “answer as many as you can in the time, you won’t finish it so don’t worry”. So you could make a point of nonchalantly handing it in with a couple of minutes to spare saying “I finished it OK….”

The only one of those that is non-trivial is no 9 (and it didn’t help on the iPhone that I had to guess the end of the question!)

But you don’t need to work it out because only one of the four options is even approximately right.

My son is in the first year of secondary school (i.e. 5 years from doing GCSE’s ) and I would be very surprised if he did not get at least 9/10.

Whilst he enjoys maths I would hope the majority of his classmates could also get well over 50%.

Therefore the idea that University students could not answer almost all the questions horrifies me.

BiS,

Those of us of a certain age, who weren’t allowed calculators in a Maths O-level but *were* allowed slide rules – which put a major premium on understanding what order of magnitude your answer should be (as well as a healthy scepticism about too many significant figures of accuracy – I’ve seen fellow students painstakingly work out that an answer is “14.342593 plus or minus 21.843875%” because if the calculator shows eight digits, it must be that precise) had to learn the “looks about right” estimate. I flashed up a calculator for the restaurant question, for example, because there were two credible options for the answer – but I could get rid of half the options at a glance.

More generally, when recruiting candidates I had to weigh A-level grades by a rule of thumb: an older candidate with BBC from the 1980s was probably as good as someone with ABBA from the late 1990s.

There was real concern in 1987-88 among those teaching subjects like A-level chemistry and physics that the new GCSEs wouldn’t adequately prepare students, and IIRC they were proved right – stuff we did at O-level is now considered A-level material, subjects we covered at A-level are now moved up into a BSc.

It runs right through the system. Field theory that I did as a fresher is now second year material, but only in an “option”. How the hell you can be a physicist without field theory beats me.

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Didn’t like the restaurant question because it said ‘exactly’, and you can’t divide 3p by 4 exactly.

My boy in y6 primary school already does all this and more.

10/10 and I worked each out before looking at the multiple choices. For many questions it would clearly have been easier to look at the choices first as knowing the approximate answer would be enough to choose the only plausible option.

The restaurant question is unrealistic: if you want to share the bill in such circumstances split the amount to a convenient round number and round up the contribution to make up the tip. ie 4×22 covers the bill, so everybody pays 23 or 24 if you like the service.

Maths tests down the years1960s: A forester sells a load of timber for £100. His cost of production is 4/5 of the price. What is his profit?1970s: A forester sells a load of timber for £100. His cost of production is 4/5 of the price, or £80. What is his profit?1980s: A forester sells a load of timber for £100. His cost of production is £80. Did he make a profit? (Answer Yes or No )1990s: A forester sells a load of timber for £100. His cost of production is £80 and his profit is £20. Underline the number 20.2000s: A forester cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands. He does this so he can make a profit of £20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the birds and squirrels feel as the logger cut down their homes? (There are no wrong answers, and if you feel like crying, it’s OK).CM

Can’t wait for the 2010’s ‘maths’ test… Perhaps by then the squirrels will have re-identified as gerbils (or hamsters), and the logger, now that she is considerate and caring, will adopt them as pets.

Calculators are allowed? That’s the shocking bit. You shouldn’t even need a pencil for these.

I can recall the absence of calculators. Apart from those mechanical wonders full of cogs you turned like a coffee grinder. I started my working life on the stock exchange calculating in shillings, pennies & farthings, to exact figures, in my head. Since then, everything I’ve done has required maths. Including navigating an aircraft. Still rarely use a calculator, although I like slide rules.

I got so low a result in my mock O-Level, small single figures, they wouldn’t let me sit the exam. Bothered, I wasn’t.

The Sage is doing something seems lost on many people today. The rough approximation. Given that, you can asses whether an answer is realistic. I’d reckon I do the same with most every calculation I make.Absolutely. This drives me nuts at work, the French just cannot do approximation. Also, very few understand the maths behind that last question regarding the £108 price tag which has already been reduced by 10%. I have supposed engineers who mastered in maths telling me that if Y is 80% of a total I need to add (Y x 20%) to reach the total.

When I did A level maths in 95/96 the teachers gave us old O level papers to practice on.You, sir, would have been sitting those exam much the same time as me.

“You never did reverse percentages in primary school. You are kidding yourself.”

Possibly not. It is taught at primary school though perhaps not at all primary schools and not to all kids. But it is a classic 11+ question so for regions with the 11+ it will certainly still be taught; moreover it is in year 6 SATS papers.

2010s: The teacher does not understand the question, and has dispensation for Dyslexia.

He conveys this to the class by interpretive dance, and proceeds with todays planned lessons regarding the relative merits of using recycled bioplastic in this weeks scrapbooking assignment about the feelings of logging-related PTSS-affected squirrels.

Johnny, however, gets a Polite Warning and a trip to the Headmaster for a Mild Admonition for looking at the paper on the teachers’ desk and attempting to answer the Questions.

A parent-teacher meeting with a Mediator is held to adress Johnny’s inability to Assimilate in the Group, and on careful consideration it is decided that Johnny should have remedial therapy to alleviate his tendency to asocial behaviour.

I could go on, but you get the picture..

This is the sort of stuff you would test an entrant to a trade course on at the local college to see if they needed a remedial course first. Of course the local colleges are all ‘Universities’ now but presumably the entrance requirement is no higher:(

My O level maths exam in 1965 included calculus – both differentiation & integration. And both were necessary for stuff we did at A level.

This isn’t good preparation for a STEM degree – it needs several more levels of difficulty, especially for anything that is based on physics, i.e. just about everything. I would also ask rather more than this for a humanities degree too, given the dire rubbish we see nowadays from journalists, pols & ‘activists’.

Raj – I hadn’t read the piece, just went straight to the test. I didn’t realise university students were included too. Yikes.

One thing I would say – I used to be a big believer in people being able to do a lot of this stuff in their heads, calculators ought to be verboten in exams etc, but a friend of mine changed my mind by simply pointing out that we will pretty much always have calculators now. (Very obvious, it just hadn’t really occurred to me in the way he said it.) They’re ubiquitous and will stay that way, I’d have thought, so maybe we don’t need to worry about doing stuff in our heads. Though I do think there’s a satisfacttion, and possibly some fringe benefits in logic etc, to be picked up that way.

(I know there are jobs which require mental arithmetic and head maths and always will, I’m on about the generality.)

@Interested– I agree that there’s little benefit in being able to multiply 4-digit numbers in your head to 7 or 8 digit accuracy, But it’s important (if you’re relying on a calculator) to at least have a feel for the order of magnitude of your expect answer and perhaps the leading digit. That way you may spot that you’ve accidentally transposed two digits or pressed – when you meant +.Don’t get me started on Excel. I remember reading a survey by one of the large accountancy firms that claimed 75% of the spreadsheets they audited contained errors, in my experience that’s a low estimate. But if it comes out of a computer it must be gold, innit.

@BiS– I started my actuarial life working with a Monroe ‘coffee grinder’. Supervisors got electric (not electronic!) models, but the real pros could ‘twiddle’ faster. Later I graduated to a programmable Epic 3000:http://www.vintagecalculators.com/html/monroe.html

I’ve been teaching my daughter about approximation, like many I did o level with no calculator and approximating was part of the process for us, also spotting known whole number solutions (Pythagoras solutions for whole numbers 3,4,5 etc.)

As for not needing mental maths as calculators are ubiquitous then that leads to the problem highlighted of writing down the answer just because that’s what the calculator said without understanding it.

Multiple choice made that test easy, though I worked most of the answers out before even looking.

I won a Hewlett Packard LED calculator (one of the non-programmable ones) in a maths competition for primary school kids in the late 1970s.

It wasn’t allowed anywhere near school once they’d taken the photograph of it.

A few years later, I was using graphing calculators in exams.

I was definitely taught some quadratic equations at primary school, simple ones, learning how to expand and contract bracketed equations, that sort of stuff. That would have been early 80s, State primary school.

@ Interested

What happens when your calculator has a flat battery and gives the wrong answer? I got talked into buying a calculator in my late 20s and a year later nearly got caught out – but the answer just looked wrong so I stopped and slowly (being badly out-of-practice) did the sum in my head.

Chris – Agree completely re estimating and orders of magnitude etc. When I did the above test I did all but the final two by looking at the answers and working out which was most likely.

John – if you work in a role which requires a calc you’ll have batteries and/or back ups. Or will be able to get them.

As I said there are roles where you will always need raw mental skills eg artillerymen need to be able to calculate in their heads, pilots need to retain ability to navigate in a non GPS world, but for most people it’s just not needed any more, or not to the extent it once was.

I used to take your view but no longer do. More important IMO for kids to get to know how to use calc than how to do it longhand.

I know some standards have slipped in recent years, particularly round mental arithmetic, but some of the commentators here are alleging that students now are

fouryears behind what they used to be!For example, doing quadratic factorising at primary school, or doing reverse percentages, dividing by 0.9, at primary school. Without calculators mind.

I’m sorry, but that is not credible. I’d like to see some evidence of these amazing allegations.

People are not stupider (actually, thanks to better food and health IQs are rising) and teaching hasn’t changed

thatmuch — there are still schools that teach effectively exactly like they did 50 years ago, mostly private.If what some of you say were true then when Britain was tested on the international PISA testing you’d not just be behind the likes of Shanghai, but you’d be on a separate page.

I think it much more likely, indeed certain, that many of you have rosy tinted visions of what was done in the past. (And having some smart kids briefly examine some extension stuff to keep them busy is not the same as it being taught at that level.)

I know for a fact, that New Zealand’s secondary school kids do not do a program vastly different from what they did 40 years ago. Actual fact, because I have the exam papers from back then.

(And don’t scoff that us colonials are behind you, because our PISA results are better. And we have more immigrants than you, so that’s no excuse either.)

What happens when your calculator has a flat battery and gives the wrong answer?What you are ignoring is opportunity cost.

If we spend years drilling high school kids on long multiplication and long division then they are not learning something else. All for a skill that they don’t need. Why not teach them Latin while you’re at it?

Which is different from not teaching them to approximate to check their answer. I teach it consistently, and they ignore me equally consistently.

While saving up for a calculator in the mid-80s I worked out how to do trig with pencil and paper – and no tables! I just remembered constants for SIN 0deg, 30deg, 60deg, 90deg, then did straight-line interpolation. eg, SIN 50 is 2/3 of the way between SIN 30 and SIN 60. Good enough for one decimal point of accuracy.

CD: I did matrices, basic trig, factoring binomials (using the polynomial remainder theorem), elementary Euclidean geometry proofs and base conversion at prep school in the 70’s/early 80’s.

All of those problems should be doable in your head. Fiddling with mental arithmetic keeps you sharp. I was walking into the supermarket at lunchtime dividing 745 by 11 in my head because I didn’t want to look like a twat by pulling out my phone. It’s obviously 67 8/11 and 8/11 is easy. If you want to multiply two digit numbers and you can remember your squares up to 99 then you can usually do difference of two squares plus a fix-up if the difference is odd. (10 x + 5)² is just 100 (x² + x + 1/4), and so on.

Now here’s a poser — if you were to answer this multiple choice question randomly, what would your chance of getting the correct answer be

A 50% B 25% C 0% D 25%

BiCR

Prep school (in my experience) goes to thirteen though (in case you’re right on the school but wrong on the year?).

@ Interested

Of course I could afford to buy a new battery – but it’s when I didn’t know that I needed one that the problem arose.

@ Chester Draws

You seem to be dismissing my point in one sentence and supporting it in the next. Approximation is what tells me if the calculator has failed or if (more likely) I’ve fat-fingered – the latest freebie calculator I was given I can depress four keys at once with a finger (six with my thumb).

Lingua latina docet fons linguae novae.

@ BiCR

That’s mildly impressive. I didn’t do matrices and I only did trig because the maths master wanted to keep the scholarship set interested (and quiet) after our exams while he coached the rest for CE.