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Mathematics examination paper from 1970 watch

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    (Original post by Shayke)
    Wow, thanks for posting.

    Really does show how the syllabus has changed. Sure, A level Maths can be difficult. But compared to 1970, its been dumbed down greatly.. =|
    Thing is, in another 40 years time someone could post a current A level Maths paper, and students would think, how the hell did anyone do that? Purely because the syllabus has changed. I'm sure if we'd learnt the 1970s syllabus we would be able to do attempt those questions!

    Did you do the edexcel C3 paper back in the summer? The Hitler one?
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    (Original post by puella_optima)
    Thing is, in another 40 years time someone could post a current A level Maths paper, and students would think, how the hell did anyone do that? Purely because the syllabus has changed. I'm sure if we'd learnt the 1970s syllabus we would be able to do attempt those questions!

    Did you do the edexcel C3 paper back in the summer? The Hitler one?
    This is very wishful thinking. I did maths, fm, and STEP over 20 years ago. I have tutored recent maths a level students. There is nothing I can see in any of the maths a level syllabi that wasn't in pre 1990. The current a level isn't really a specialist maths qualification. It is a general maths qualification for scientists. Most mathematics university courses need at least a term of catch up to cover missed material and the level of rigour required for a maths degree. Anyone who did an a level pre the big dumb down just looks a what is now called maths just feels sad. If they are a bit cruel and superior then they laugh. Anyone talented is being cheated. Others are deluding themselves.
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    This is my subbing post. I'll have a go at this paper later on.

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    (Original post by lifelonged)
    This is very wishful thinking. I did maths, fm, and STEP over 20 years ago. I have tutored recent maths a level students. There is nothing I can see in any of the maths a level syllabi that wasn't in pre 1990. The current a level isn't really a specialist maths qualification. It is a general maths qualification for scientists. Most mathematics university courses need at least a term of catch up to cover missed material and the level of rigour required for a maths degree. Anyone who did an a level pre the big dumb down just looks a what is now called maths just feels sad. If they are a bit cruel and superior then they laugh. Anyone talented is being cheated. Others are deluding themselves.
    'The big dumb down'? That's a little harsh.. Personally, I was expecting A Level maths to be more difficult but to say it's pretty easy is not reasonable, most people who are talented do Further Maths too. I am not a scientist at all so for me, maths isn't a 'general qualification'.

    A lot of students getting B, A or A* at GCSE don't carry on to A Level because they don't think they're good enough. I got an A* at GCSE and still went into AS with no confidence. To say maths is easy today is unfair for current candidates
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    (Original post by puella_optima)
    'The big dumb down'? That's a little harsh.. Personally, I was expecting A Level maths to be more difficult but to say it's pretty easy is not reasonable, most people who are talented do Further Maths too. I am not a scientist at all so for me, maths isn't a 'general qualification'.

    A lot of students getting B, A or A* at GCSE don't carry on to A Level because they don't think they're good enough. I got an A* at GCSE and still went into AS with no confidence. To say maths is easy today is unfair for current candidates
    Dumbing down does not imply "easy" it suggests "easier"

    GCSE maths is easier than O'level was and A'level is easier than it used to be

    That does not indicate that current Sixth Formers try less than we did, they have to make a similar shift from their Level 2 to their Level 3
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    (Original post by puella_optima)
    'The big dumb down'? That's a little harsh.. Personally, I was expecting A Level maths to be more difficult but to say it's pretty easy is not reasonable, most people who are talented do Further Maths too. I am not a scientist at all so for me, maths isn't a 'general qualification'.

    A lot of students getting B, A or A* at GCSE don't carry on to A Level because they don't think they're good enough. I got an A* at GCSE and still went into AS with no confidence. To say maths is easy today is unfair for current candidates
    It's definitely easier!

    GCSE maths contains about half the content of my old Maths O Level. Moreover, the grade C threshold for Higher Tier GCSE seems to be about 26% whereas it was something like 40 or 45% for O level. You can draw your own conclusions about how grades compare today with then.

    At A level there is a similar inflation of grades. With the old pre-modular A levels you had to be "A grade standard" in the final year's work in order to receive an A grade overall, Now I see people who are getting D grades in their A2 exams who have accumulated so many marks from AS levels that they still get awarded a final grade A.

    These are just a couple of specific comparisons I can think of, but there are even more worrying trends that can be observed on TSR - the number of people taking A level Maths (or even Maths degrees) who cannot handle basic fractions or indices is quite shocking!
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    (Original post by puella_optima)
    'The big dumb down'? That's a little harsh.. Personally, I was expecting A Level maths to be more difficult but to say it's pretty easy is not reasonable, most people who are talented do Further Maths too. I am not a scientist at all so for me, maths isn't a 'general qualification'.

    A lot of students getting B, A or A* at GCSE don't carry on to A Level because they don't think they're good enough. I got an A* at GCSE and still went into AS with no confidence. To say maths is easy today is unfair for current candidates
    I'm not making any statement about the current candidates, other than they are being cheated out of a high quality educational experience in a great subject. I don't think that people are any less intelligent (I also don't think that there are any more intelligent - even with Flynn effect taken into account). At least with the old O level (and even the first GCSE), A level, S level, STEP students had an opportunity to develop as mathematicians. I can't see how the current A level prepares you for degree level mathematics (or much else really). If the politics was taken out of it and educators carried on doing what they do well then maybe things would be different. You'd have to accept that grades wouldn't improve every year, no multiple retakes, and some people would fail (and have to retake at the end of the next year). The current A level maths is a mockery and very capable mathematicians are being short changed. It used to be that getting a C or D at A level maths was fairly respectable, now this is seen as a poor result and wouldn't get you anywhere.

    I'm afraid that as the other posters have pointed out, maths A level is much easier (compared to the old exam) now. Why:

    i) Lots of content removed, very little rigour and proof. The syllabus is becoming comparable with the early GCSE, and is very similar to O level now.
    ii) Questions have become a little spoon fed. Do this and this, then this. The old style (as you can see from the samples) were open and forced you to use a mathematical method of attacking and solving a problem - often more than one way to arrive at the correct answer.
    iii) All these modules with the option of retakes (in many cases) don't allow for real joined-up thinking. Much interesting mathematical thought involves making links with previously unrelated areas e.g. look at Andrew Wiles' proof of Fermat's Last Theorem, beautiful, and links very unrelated branches of mathematics. You can do this (on a much simpler level) at A level e.g. solve mechanics with Lagrange's equations, basic Galois theory. I get lost with talk of C1 etc. there used to be just maths and a few papers to test you on it.
    iv) ...... many more silly changes........

    Puella_optima I can understand that you may feel a bit attacked by some of the themes in this thread. I would if I were a current student or teacher. A levels were designed as the top academic school level education, ideally designed to prepare for a degree. Maths is one of the oldest degrees, hence the maturity and difficulty of the A level. You used to have to attain an A in many schools to progress to an A level (or have a good reason for not getting an A in the exam). You would have been predicted to struggle at A level otherwise. A level physics students had to study maths too. All this has changed, and sadly not for the better. The only thing that is harsh to current candidates is the specific (more than many other A levels) bundling maths A level into a political game and cheating students from an education.

    If I were sitting A level maths now and I wanted to progress to a mathematical degree, I would get a good text book (most that have a heritage over 30 years and have a more modern style should do), and I would do STEP questions (or some of the other quals quoted AEA, but I don't know these), or old A level papers. If you do this then you'll have something of value and you'll (eventually) enjoy yourself much more than you would with the current A level. You'll also be one of the best prepared students on your first day of your maths degree.
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    I don't think that anyone has had a go at 10? I had an attempt below. Sorry for the brevity, had some problems with the Latex preview and it got a bit long. Can fill in any gaps if required. I can't see an alternative to implicit functions and partial differentiation. I remember partial differentiation on FM in 1991, but not implicit functions. Could be that I've blurred this with University, or could be that the 1970s S papers were really tough and that dumbing down has been going on for some time. They are very different in style to STEP. Not sure which would have been harder. Interesting thread.
    

(x+y)^3 = 9xy

So g(x,y) = 9xy - (x+y)^3 = 0



\partial g/\partial x = 9y - 3(x+y)^2





\partial g / \partial y = 9x - 3(x+y)^2

 



and by implicit differentiation:



dy/dx = - (\partial g / \partial x) / (\partial g / \partial y) = \frac{9y - 3(x+y)^2}{9x - 3(x_y)^2}



Using (x+y)^2 = \frac{9xy}{x+y}



dy / dx = (3*9xy-9y(x+y)) / (9x(x+y) - 3*9xy)



dy / dx = (2xy-y^2) /  (x^2 - 2xy) so at (h,k)



dy / dx= (2hk-k^2) /  (h^2 - 2hk) as required

    The tangents parallel to the x axis fall out:
    dy / dx = 0, gives (2/3, 4/3) i.e. y=4/3 and there is another tangent parallel to the x axis at (0,0) i.e. y=0 (but derivative not defined).

    So the end points of this chord are given by the solutions to:
    

y+x=a (for some constant a - this line is tangent to y+x=0) i.e .y = -x +a



and (x+y)^3 = 9xy



sub in gives a quadratic 9x^2 - 9ax +a^3 = 0



Solution (by formula) x= (3a \pm a \sqrt{9 - 4a})/6



and y = -x + a (as above)



The mid point is the arithmetic average of these two points i.e.



(a/2, a/2)



So the locus is y=x, but it is bounded, from the discriminant of the quadratic 9 - 4a \ge 0 \Rightarrow a \le 9/4 i.e. bounded at point (9/8, 9/8).
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    that looks really hard
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    For 1) the discriminant of any normal form cubic of the form Ax^{3}+Bx^{2}+Cx+D is:

    (BC)^{2}-4(AC^{3}+B^{3}D)+18ABCD-27(AD)^2

    For all real (and distinct) roots, this has to be >=0 (for real and equal, ...=0)

    here, since A=1, B=0 C=a, D=b we get:

    108a^{3}-27b^2 \geq 0....

    (P>S> - I`m Scottish - so please forgive my ignorance of the English curriculum if this isn`t covered) - (and indeed my post!)
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    (Original post by lifelonged)
    I don't think that anyone has had a go at 10? I had an attempt below. Sorry for the brevity, had some problems with the Latex preview and it got a bit long. Can fill in any gaps if required. I can't see an alternative to implicit functions and partial differentiation. I remember partial differentiation on FM in 1991, but not implicit functions. Could be that I've blurred this with University, or could be that the 1970s S papers were really tough and that dumbing down has been going on for some time. They are very different in style to STEP. Not sure which would have been harder. Interesting thread.
    

(x+y)^3 = 9xy

So g(x,y) = 9xy - (x+y)^3 = 0



\partial g/\partial x = 9y - 3(x+y)^2





\partial g / \partial y = 9x - 3(x+y)^2

 



and by implicit differentiation:



dy/dx = - (\partial g / \partial x) / (\partial g / \partial y) = \frac{9y - 3(x+y)^2}{9x - 3(x_y)^2}



Using (x+y)^2 = \frac{9xy}{x+y}



dy / dx = (3*9xy-9y(x+y)) / (9x(x+y) - 3*9xy)



dy / dx = (2xy-y^2) /  (x^2 - 2xy) so at (h,k)



dy / dx= (2hk-k^2) /  (h^2 - 2hk) as required

    The tangents parallel to the x axis fall out:
    dy / dx = 0, gives (2/3, 4/3) i.e. y=4/3 and there is another tangent parallel to the x axis at (0,0) i.e. y=0 (but derivative not defined).

    So the end points of this chord are given by the solutions to:
    

y+x=a (for some constant a - this line is tangent to y+x=0) i.e .y = -x +a



and (x+y)^3 = 9xy



sub in gives a quadratic 9x^2 - 9ax +a^3 = 0



Solution (by formula) x= (3a \pm a \sqrt{9 - 4a})/6



and y = -x + a (as above)



The mid point is the arithmetic average of these two points i.e.



(a/2, a/2)



So the locus is y=x, but it is bounded, from the discriminant of the quadratic 9 - 4a \ge 0 \Rightarrow a \le 9/4 i.e. bounded at point (9/8, 9/8).
    In fact students who have covered C4 should know how to find dy/dx here but they don't learn about the implicit function theorem.
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    Where you have found the examination paper, Mr M?

    The examination of 1970 is very difficult compared to the requirements today. In my opinion at least. I cannot remember me to have had such difficult tasks in mathematics.
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    (Original post by BabyMaths)
    In fact students who have covered C4 should know how to find dy/dx here but they don't learn about the implicit function theorem.
    OK, it looks like C4 is part of what used to be FM? I had a look at one syllabus (EdExcel) it does look like implicit differentiation is on there. The syllabi do seem to vary quite a bit. It must make the first term of University maths interesting. I think that it used to be a case of the non FM A level students having a term to catch up.

    I suppose the bit that interests me as a father of a young girl who is interested in maths is what would you do to get to a good standard at age 18. It looks like STEP preparation is a good way to go. It is a shame that some of the ideas about proof have been taken out of A level maths.
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    Damn that's hard compared to A Level maths.

    One time my A Level physics teacher gave me an O Level physics paper from the '70s and it made GCSE physics look like a joke.
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    (Original post by lifelonged)
    I suppose the bit that interests me as a father of a young girl who is interested in maths is what would you do to get to a good standard at age 18. It looks like STEP preparation is a good way to go. It is a shame that some of the ideas about proof have been taken out of A level maths.
    The two traditional Bostock and Chandler books provide pretty good grounding:
    Mathematics: The Core Course for A Level
    Further Pure Mathematics (written with C Rourke)

    They're not perfect, and of course the topics won't be in any order that corresponds to a modern A level course, but if your daughter can master those 2 books then she won't go far wrong!

    Applied Maths is a different kettle of fish - I would stay away from Decision Modules since they do nothing to develop students' mathematical abilities and focus on Mechanics and Statistics. The sad thing about Statistics these days is that the S1 module contains such awful, mind-numbing tedium that it can't be taught in any interesting way and tends to put people off stats before they can even get into more challenging ideas
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    (Original post by alow)
    Damn that's hard compared to A Level maths. (...)
    Good to know that members like you feel the same.
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    (Original post by davros)
    The two traditional Bostock and Chandler books provide pretty good grounding:
    Mathematics: The Core Course for A Level
    Further Pure Mathematics (written with C Rourke)

    They're not perfect, and of course the topics won't be in any order that corresponds to a modern A level course, but if your daughter can master those 2 books then she won't go far wrong!

    Applied Maths is a different kettle of fish - I would stay away from Decision Modules since they do nothing to develop students' mathematical abilities and focus on Mechanics and Statistics. The sad thing about Statistics these days is that the S1 module contains such awful, mind-numbing tedium that it can't be taught in any interesting way and tends to put people off stats before they can even get into more challenging ideas
    Thanks Davros, I think that I still have the Bostock & Chandler books from 1990. Took me two months of paper rounds to save up for. They were ok, but I still felt that they took a bit of a for dummies approach. I had to dig out my fathers old Schaum books to answer some of my questions. There was (and maybe still is) a bit of a big gap between A level books and the books that you'd use for 1st term degree maths. Maybe we treat A level students as though they can't handle difficult concepts? I'll have to dig out B&C and refresh my memory.

    I agree with you about stats. It was pretty bad 25 years ago. Nothing interesting like the foundations of probability and the theoretical underpinnings. Shame as I use the subject a lot now, but I ignored it at A level.
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    (Original post by Kallisto)
    Where you have found the examination paper, Mr M?
    I think I got it from Edexcel but can't really remember. It was 4 years ago!
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    (Original post by Mr M)
    I think I got it from Edexcel but can't really remember. It was 4 years ago!
    G has quite a few in the emporium
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    (Original post by TenOfThem)
    G has quite a few in the emporium
    Pretty sure that's where I found it.
 
 
 
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