On 24-9-2009 12:24, Warp wrote:
> Invisible <voi### [at] devnull> wrote:
>> Well, I suppose you could argue that, in the real world, if you want to 
>> do a nontrivial calculation and you actually need an exact answer, 
>> you'll use some sort of computation device.
> 
>   But that's not the point. "Just use a calculator" is not didactic, it
> doesn't teach anything, and will only make people lazy. This will bite
> back in the future, when people won't become programmers because they
> lack the proper education and way of thinking.

agreed, see below.

>>>   The same is true for other mathematical operations, such as division
>>> and square root.
> 
>> Out of curiosity... how the hell *do* you actually calculate the square 
>> root of something? I've always wondered.
> 
>   Your lack of google skills don't surprise me. Slightly more surprising is
> that someone of your age hasn't been taught that at school.

I have recently discussed this with a couple of people at the places I 
work. It seems teaching doing a square root by hand was removed from the 
basic training about 40-50 years ago. It wasn't in my training anywhere, 
though I do know it after having sort of derived it myself. (That is of 
course why it is a particular favourite of mine).
I am now teaching digital technology to first years electronic 
engineering at an applied university (whatever that is in English). One 
of my colleagues that is a few years older and is teaching VHDL and that 
sort of stuff actually did not know it. Someone from my department in 
the hospital did know and he is about the same age. I wouldn't be 
surprised if the only reason he learned it was that his teacher liked it 
and not that it was in the official end terms. (BTW I am 47 and they are 
begin-mid 50s).

More disturbingly: I noticed that some of my students did not know long 
division. I haven't asked for long multiplication (yet). It seems to 
have dropped from the program for the same reasons as you mention for 
the US. The teachers don't understand it and don't see why an innocent 
kid needs to know such a overly complicated algorithm if they can use a 
pocket calculator. To get things in perspective: the majority of people 
training for teacher fail the end test for our 12 year olds in maths.

One of the first courses at this university is in basic maths. Which is 
simply trying to get them at the level of the end terms of the school 
they have just finished (actually even less than that). We do start with 
addition, subtraction, long multiplications and divisions, and fractions...

>
http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Digit_by_digit_calculation

>>>   They are missing the point. By teaching lazy rules of thumb, they are
>>> failing to teach a higher concept: Algorithmical thinking. You can't
>>> program a computer with just "rules of thumb".
> 
>> Well, most people won't ever need to be computer programmers. It _is_ a 
>> minority occupation. Indeed, most professions don't really involve 
>> higher mathematics in any way.
> 
>   Then you wonder why some countries are more proficient than others.
> It's probably a common trait in the more proficient countries that the
> schooling system is *not* driven by a "only very few people will ever
> need these skills, thus we'll just skip teaching them" ideology.

The common fear with those in teaching maths, physics and chemistry in 
the Netherlands  is that we will be with the countries at the bottom in 
a few years/decades. We are still doing OK but when the current 
generation retires we are left with an incompetent mathophobe group of 
teachers.
 From my personal experience I know that also the writing skills are far 
below what was thought in my days.

OTOH, every generation will say so.

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On 09/24/09 14:14, andrel wrote:
>> Your lack of google skills don't surprise me. Slightly more surprising is
>> that someone of your age hasn't been taught that at school.
>
> I have recently discussed this with a couple of people at the places I
> work. It seems teaching doing a square root by hand was removed from the
> basic training about 40-50 years ago. It wasn't in my training anywhere,

	I doubt it was ever in basic training universally. Probably only in 
some countries. I know folks much older who know about it - seeing it at 
some point in their lives, but never were formally taught it.

	In my text book (7th or 8th grade), it was there in one of those 
special "fun" sections, and it's possible that the teacher even made us 
do a few problems using it, but it wasn't required to know, and I never 
memorized it.

> More disturbingly: I noticed that some of my students did not know long
> division. I haven't asked for long multiplication (yet). It seems to
> have dropped from the program for the same reasons as you mention for
> the US. The teachers don't understand it and don't see why an innocent

	That's *really* bad. Long division is not something that can be 
replaced with calculators. You use it for algebraic expressions as well. 
Unless, of course, it's degraded to the point where you don't need 
algebra and you just let Maple do everything.

	I know a lot of people criticize some of the "new math". I'm not that 
certain in either direction. I find long multiplication and long 
division to be good for computing stuff fast, but gives virtually *no* 
insight into anything. The techniques in "new math" actually give you a 
fundamental understanding of what's going on, and has more "right" to be 
called mathematics. But it's likely not practical to do it that way each 
time.

	If I had kids, I'd try to make sure they learn both. I consider either 
approach alone to be seriously flawed.

	Reminds me of a bunch of math graduate students I've known. Fairly slow 
at doing integrals, etc. But a physics student or a good engineering 
student is much faster at doing them. It's because the math students are 
not concerned with calculations (for the most part) - but the others are.

	In any case, my advisor in grad school had a no-calculators policy for 
exams. And this was for engineering courses, where you have to calculate 
stuff with real numbers, and not just symbolically. I always felt that 
if I were a professor, I'd do likewise for exams. I wonder how long 
before such professors are shunned by their own colleagues.

> One of the first courses at this university is in basic maths. Which is
> simply trying to get them at the level of the end terms of the school
> they have just finished (actually even less than that). We do start with
> addition, subtraction, long multiplications and divisions, and fractions...

	I often wonder. A lot of people (including myself sometimes) feel that, 
say, mathematics education is degrading over time. In my undergrad 
institution, a few decades ago, the lowest math class was introductory 
calculus. As the years went by, they needed to put a remedial precalc 
course that was strictly not for credit. Then as time went by, they 
converted that to a proper course with credits, and put a remedial 
algebra course for no credits. Now that course is offered for credit.

	Likewise, some of the things they often teach in the first year of 
graduate school here in mathematics is often taught in the 4th year of 
undergrad in universities in other countries (I just saw one where it's 
taught in the 3rd year of undergrad).

	Is the math education at high school etc degrading, requiring all these 
remedial courses?

	Possibly.

	Another explanation is that as a percentage (and of course, in absolute 
numbers) of the population, more and more students are getting educated 
and graduating high school, and the education system is having trouble 
keeping up. Put another way, the percentage of people age 18 in this 
country who have a "solid" background in mathematics may have actually 
stayed the same (not gone worse) - it's just that the percentage 
graduating and moving on to university is higher (and not just because 
standards have gone down).

	I used to be firmly in the first camp, but I really don't have the 
data, and it wouldn't surprise me if the second reason is the bigger one.

> The common fear with those in teaching maths, physics and chemistry in
> the Netherlands is that we will be with the countries at the bottom in a
> few years/decades. We are still doing OK but when the current generation
> retires we are left with an incompetent mathophobe group of teachers.
>  From my personal experience I know that also the writing skills are far
> below what was thought in my days.
>
> OTOH, every generation will say so.

	I wonder how much of this really matters.

	Take a number of east European countries. And Russia. When it comes to 
mathematics, they used to (and probably still do) beat, say, the US 
handily. Not just the brilliant ones, but also the average high 
school/university graduate.

	In many technical areas (except perhaps engineering), the US has been 
consistently behind those and other countries for most of the 20th century.

	Yet it doesn't hurt the country much.

	I think what matters is maintaining a fairly good average (not just of 
students, but the whole population). I don't really know, but I suspect 
those countries tended to be elitist. Only the really good got to go to 
universities, whereas the rest went to simple trade schools. And that 
results in an overall lower average. An elite group of a few brilliant 
people cannot impact the country as much as a huge group of above 
average folks.

	I think the UK used to be like that and they probably changed it some 
decades ago. I've met a bunch of older people from there who couldn't 
get into university to become an engineer, but in terms of their 
abilities and grades, had they lived in the US, they would have.

-- 
If you think nobody cares, try missing a couple of payments.

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Neeum Zawan <m.n### [at] ieeeorg> wrote:
>  Reminds me of a bunch of math graduate students I've known. Fairly slow
> at doing integrals, etc. But a physics student or a good engineering
> student is much faster at doing them. It's because the math students are
> not concerned with calculations (for the most part) - but the others are.

From my PDE's textbook:

"...Our efforts will be largely devoted to proving mathematically the existence
of solutions. ... This may seem like wasted or misguided effort, but in fact
mathematicians are like theologians: we regard existence as the prime attribute
of what we study.  But unlike theologians, we need not always rely upon faith
alone."

>  In any case, my advisor in grad school had a no-calculators policy for
> exams. And this was for engineering courses, where you have to calculate
> stuff with real numbers, and not just symbolically. I always felt that
> if I were a professor, I'd do likewise for exams. I wonder how long
> before such professors are shunned by their own colleagues.

The professors I thought the most highly of were the ones who basically said,
"Open book, open notes, calculators allowed--but it won't help."  I really do
think estimation is an undervalued skill though.  The ability to stand there,
figure out orders of magnitude in your head, and come up with some quick
approximations can save a lot of time blindly pursuing totally inappropriate
methods.  That sounds obvious, but it's really remarkable how completely some
people lack basic problem-solving and analysis skills.

>  I often wonder. A lot of people (including myself sometimes) feel that,
> say, mathematics education is degrading over time. In my undergrad
> institution, a few decades ago, the lowest math class was introductory
> calculus. As the years went by, they needed to put a remedial precalc
> course that was strictly not for credit. Then as time went by, they
> converted that to a proper course with credits, and put a remedial
> algebra course for no credits. Now that course is offered for credit.

Maybe some people need a more remedial course, so they add a remedial course,
just to get everyone up to speed.  Suddenly, the bottom rung of the ladder isn't
the bottom anymore, so all the people who were struggling in precalc move up a
step.  And if you're not falling off the bottom, why worry?  People just expand
to fit the container, in a sense.

>  Likewise, some of the things they often teach in the first year of
> graduate school here in mathematics is often taught in the 4th year of
> undergrad in universities in other countries (I just saw one where it's
> taught in the 3rd year of undergrad).

"it" being what?  Just curious...

>  Another explanation is that as a percentage (and of course, in absolute
> numbers) of the population, more and more students are getting educated
> and graduating high school, and the education system is having trouble
> keeping up. Put another way, the percentage of people age 18 in this
> country who have a "solid" background in mathematics may have actually
> stayed the same (not gone worse) - it's just that the percentage
> graduating and moving on to university is higher (and not just because
> standards have gone down).

This actually seems plausible.  Still, they choose their program, so to allow
more people through, the standards must be relaxed.  As I said above, relaxed
standards in turn reduce effort put forth by students, compounding the problem.
Just a guess, of course.

>  I think what matters is maintaining a fairly good average (not just of
> students, but the whole population).

Let's hear it for mediocrity!

 - Ricky

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> I have recently discussed this with a couple of people at the places I 
> work. It seems teaching doing a square root by hand was removed from the 
> basic training about 40-50 years ago.

I think this was the case in the UK too, my maths teacher mentioned it as 
something they had to do "in the old days", but it certainly wasn't on the 
curriculum.  Of course long multiplication and long division were, also 
later on in school we had to do long division with algebraic expressions... 
fun!

> To get things in perspective: the majority of people training for teacher 
> fail the end test for our 12 year olds in maths.

"Those that can't do, teach, and those that can't teach... teach gym."

> One of the first courses at this university is in basic maths. Which is 
> simply trying to get them at the level of the end terms of the school they 
> have just finished (actually even less than that). We do start with 
> addition, subtraction, long multiplications and divisions, and 
> fractions...

Hmm at my University they send us a booklet over the summer holiday before 
we started, basically telling us we should know all of this and to learn the 
rest before you come.  It was more to just fill in the gaps between the 
different curriculums at different schools, but mostly we had covered it 
all.

> From my personal experience I know that also the writing skills are far 
> below what was thought in my days.

What makes it worse in the UK is that every year the media congratulates 
everyone with record pass rates for school exams.  You simply have to 
compare a maths exam paper (or even text book) with one from 20 years ago to 
realise how massively easier it is today.  Stuff that was in my A-level test 
(taken when you're 18) was the same stuff in my dad's O-level test (when 
you're 16).

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"triple_r" <nomail@nomail> wrote:
> I really do
> think estimation is an undervalued skill though.  The ability to stand there,
> figure out orders of magnitude in your head, and come up with some quick
> approximations can save a lot of time blindly pursuing totally inappropriate
> methods.  That sounds obvious, but it's really remarkable how completely some
> people lack basic problem-solving and analysis skills.

My physics course at university included a weekly 'workshop' slot in the
teaching. Among other things, this focused on estimating order-of-magnitude
answers to complicated yet unspecific problems, using nothing but basic physics
formulae from memory, sensible guesses for data and paper & pencil for
calculation. Questions like 'how much water might be vaporised if an asteroid
hit the ocean?'

In hindsight, it was probably one of the most valuable skills I was taught in
that degree programme.

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scott wrote:

> I think this was the case in the UK too, my maths teacher mentioned it 
> as something they had to do "in the old days", but it certainly wasn't 
> on the curriculum.  Of course long multiplication and long division 
> were, also later on in school we had to do long division with algebraic 
> expressions... fun!

I envy schools where they actually *teach* algebra.

Everything I know about algebra is what little I've been able to figure 
out for myself and scrape out of library books.

> "Those that can't do, teach, and those that can't teach... teach gym."

Incidentally, our gym teacher was the most terrifying guy in the school. 
Very fair, but... DO NOT get caught doing bad stuff! My God, the horror...

> What makes it worse in the UK is that every year the media congratulates 
> everyone with record pass rates for school exams.

What happens is that every single year, the pass rate either goes up, or 
it goes down. It is a statistical fact that it is almost impossible for 
the pass rate to remain exactly the same.

If it goes up, even slightly, even by a statistically insignificant 
amount, the media starts screaming about "exams are getting too easy". 
Not, you know, that students are getting brighter or working harder or 
new teaching methods or anything like that. No, exams are getting easier.

If it goes down, even by a statistically insignificant amount, the media 
shrieks "schools are failing our children". Not, you know, that the 
exams are getting harder, or that this was just a random fluctuation, 
but that schools are failing us.

No matter what happens, the media will yell and shout about the terrible 
moral decay of either failing schools or watered down exams. There is 
apparently nothing the education system can do about this.

> You simply have to 
> compare a maths exam paper (or even text book) with one from 20 years 
> ago to realise how massively easier it is today.

I haven't had the opportunity to try that, so I really couldn't say.

When I did my exams, I didn't think they were especially easy. And I was 
entered for the lower grades. (E.g., I got a B in science, and B is the 
highest possible mark for the paper I took.)

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> Everything I know about algebra is what little I've been able to figure 
> out for myself and scrape out of library books.

Do a maths evening class?  You should sail through the GCSE, the A-level 
might take a little bit of work to fill in the gaps of your knowledge.  For 
even further the Open University probably offer something.

> What happens is that every single year, the pass rate either goes up, or 
> it goes down.

There is a strong upward trend though in the results:

http://2.bp.blogspot.com/_mJmwQtPmusk/Rt8ZqfEnvyI/AAAAAAAAA2U/TIGBotspDds/s400/gcse-scores--ns-2007.jpg

> I haven't had the opportunity to try that, so I really couldn't say.

Maybe one of your parents has an old text book from school? I also got my 
dad's A level physics text book, and it went in to so much more detail about 
*everything* that we did in A level physics, if I wanted to read further 
about anything in school I used his books.

> When I did my exams, I didn't think they were especially easy.

I'm not talking about individuals, I'm talking about the entire population. 
When something stupid like 40% gets an A grade, how is anyone meant to tell 
the difference between the hard working kids and the really clever ones? 
They introduced the A* grade at GCSE, but the % of people getting A* has 
been steadily rising.  How are Universities meant to choose the best 
students when more and more are applying with the highest possible grades?

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>> Everything I know about algebra is what little I've been able to 
>> figure out for myself and scrape out of library books.
> 
> Do a maths evening class?  You should sail through the GCSE, the A-level 
> might take a little bit of work to fill in the gaps of your knowledge.  
> For even further the Open University probably offer something.

My mum suggested I do a maths A-level. Personally, I'm not sure where 
I'd find the money or the time. Nice idea though...

>> What happens is that every single year, the pass rate either goes up, 
>> or it goes down.
> 
> There is a strong upward trend though in the results:

Is this a fact, or just hype? It's hard to tell just by listening to the 
news. (If you believed them, bird flu is a deadly world-wide pandemic 
which is killing everything in sight...)

>
http://2.bp.blogspot.com/_mJmwQtPmusk/Rt8ZqfEnvyI/AAAAAAAAA2U/TIGBotspDds/s400/gcse-scores--ns-2007.jpg


This isn't grades, it's point scores. You can get a higher GSCE point 
score by taking more subjects.

>> I haven't had the opportunity to try that, so I really couldn't say.
> 
> Maybe one of your parents has an old text book from school? I also got 
> my dad's A level physics text book, and it went in to so much more 
> detail about *everything* that we did in A level physics, if I wanted to 
> read further about anything in school I used his books.

I have my mum's O-level maths books. (Plural.) And her logarithm tables. 
(They're very brown and crumbly.) The books talk about a lot of stuff I 
either already know or don't understand. I guess because I don't know 
the stuff that's supposed to come *before* an O-level. The books talk a 
lot about trigonometry and triangle rules; damn, at my school, maths 
classes didn't even mention triangles!

> When something stupid like 40% gets an A grade

It's that high??

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> Is this a fact, or just hype? It's hard to tell just by listening to the 
> news. (If you believed them, bird flu is a deadly world-wide pandemic 
> which is killing everything in sight...)

Or that HIV had just been cured because 51 got HIV who took the vaccine 
compared to 74 otherwise.

Anyway, I'm sure you can actually look up the real GCSE pass rates rather 
than simply reading the text of news reports.  The bbc site has a graph on 
this page:

http://news.bbc.co.uk/2/hi/uk_news/education/4202327.stm

>> When something stupid like 40% gets an A grade
>
> It's that high??

I was just making that bit up, after checking it's actually 20%.

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>> Is this a fact, or just hype? It's hard to tell just by listening to 
>> the news. (If you believed them, bird flu is a deadly world-wide 
>> pandemic which is killing everything in sight...)
> 
> Or that HIV had just been cured because 51 got HIV who took the vaccine 
> compared to 74 otherwise.

Or that, yes.

> Anyway, I'm sure you can actually look up the real GCSE pass rates 
> rather than simply reading the text of news reports.

Somewhere somebody probably has all sorts of interesting statistics. 
Damned if I know how to find them though...

>>> When something stupid like 40% gets an A grade
>>
>> It's that high??
> 
> I was just making that bit up, after checking it's actually 20%.

Hmm, I take your point. That *is* pretty high. I was expecting A grades 
to be nearer to 0.02%. (Let's face it, most people are very, very stupid.)

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