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Looking for Mathematics past papers from around 1920s onwards

Last post 13/01/11 at 17:45 by autismuk, 274 replies
Post started by intuitionist1 on 02/01/11 at 00:40

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    Posted by: intuitionist1 02/01/2011 at 00:40
    Joined on 10/11/2008
    Posts 44

    Hi,

    I am wondering whether anyone can help.

    I am trying to collect past mathematics papers - School & Higher Certificate, and GCE O, AO & A Level, in Mathematics and Further Mathematics (including scholarship papers), especially for the Oxford & Cambridge boards, during the period from the 1920s through to the 1980s. I am particularly interested in bound sets covering a number of years. Cambridge STEP papers would also be of interest.

    If you happen to have any of these available to donate, lend, or sell in softcopy or hardcopy form, I would greatly appreciate it.

    Best wishes,

    Sabbir Rahman

    BA Hons (Oxon), PhD (MIT)
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    Posted by: Betamale 02/01/2011 at 05:32
    Joined on 31/07/2010
    Posts 519

    A member here has some on his site below:

    http://www.emaths.co.uk/cms/index.php?option=com_content&view=article&id=55:very-old-exam-papers&catid=39:old-exam-papers&Itemid=57

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    Posted by: DM 02/01/2011 at 11:44
    Joined on 12/05/2003
    Posts 5,447

    Could you explain why you need them Sabbir?   All recent years are already widely available, particularly the complete set of STEP papers.  

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    Posted by: intuitionist1 03/01/2011 at 04:43
    Joined on 10/11/2008
    Posts 44

    Hi,

    I am fascinated by the way the teaching of mathematics has evolved over the years, and would like to make use of past teaching and examining materials to attempt to recreate a structured syllabus which turns back the clock on the progressive "dumbing down" that we have been, and continue to be, subjected to in recent years.

    For example, I recall thinking, back in 1986-88 when I was revising for my A levels, how simple the questions in the exams that I took then had become compared with questions that had appeared in past papers (I used to have books of past papers going back to around 1968, and during my revision I think I must have attempted pretty much every question on the further maths papers - I wish now that I had kept both the question papers and my solutions).

    It was also clear that, despite the actual syllabus not having changed a great deal over the period, the questions themselves had become much easier over time. The pre-1970 questions were relatively challenging from a problem-solving perspective and expected/required significant stamina and dexterity in algebraic manipulation, whereas the 1980s questions tended to be much shorter and straightforward and generally required less forethought to solve. To give you an idea of the difference, the average time it would take me to solve a 1980s question was around 3-4 minutes as compared with around 20-25 minutes per question on the pre-1970s papers. I also recall that the syllabus was chopped considerably the year after I took my A levels. (This was also true of the O level syllabus, which was subsequently merged with CSEs to be replaced by GCSEs).

    The general decline seems to have continued ever since then. I now teach A level mathematics on a voluntary basis on weekends, and I have to say that I am rather dismayed by the types of questions asked in today's exam papers, where almost no problem-solving ability is required and the questions are so "dumbed-down" that students need only directly apply a standard set of basic techniques which they will have memorised in advance. There is very little to distinguish the extremely bright from the merely competent, but more importantly, students are no longer being taught 'mathematics' as I understand it, but rather, how to pass a specific mathematics exam where the questions to be asked are so straightforward as to be effectively known in advance. [If students find today's A levels challenging, it is because they are so much less well prepared for them than they would have been in the past, and find themselves struggling to learn much more far too late in the day].

    It concerns me that my own children (I have three sons, aged 11, 10 and 6) are quite possibly going to be come out of school with top grades yet only a relatively superficial understanding of the subject matter.

    Coming back to the original question, I have gathered a collection of older mathematics textbooks, particularly those by C V Durell et al from the late 1920s onwards. These were dense thoughtful texts and well-planned graduated examples and numerous exercises and drills that were designed not only to ensure that the basics were clearly understood, but also to challenge the very brightest. I do not think that it would be an exaggeration to say that the standard of today's A levels seem to be roughly comparable to the basic knowledge an average 13 or 14-year old might have expected to have back in those days.

    I realise that obtaining a decent education was much more of a privilege in the distant past. However in making it more accessible it seems also to have been watered down considerably. I do not believe that children are fundamentally any less capable or intelligent than they were in the past. However it is clear that they are not being suffficiently pushed or challenged and end up achieving (and worse, expected to achieve) much less than they would have been back then. Not only are students less well prepared, so of course are the teachers that teach them, leading to something of a vicious circle. Not that any of us would like to admit as such, but those old enough to remember, or those bothering to look at past textbooks or exam papers in an objective way, will find it difficult, I would hope, to reach any other conclusion.

    I believe that each student should be given the opportunity to achieve his/her potential, and that this should not be a function of any particular syllabus put together by the government or an examinations board with any agenda which may not be geared to what is best for the students themselves. By examining how mathematics was taught in the past as well, it should be possible to rearrange topics in a structure which allows students to progress each according to their own ability. The better students should be comfortably able to take their A levels in further mathematics at 13 or 14 and should not be discouraged in any way from doing so. Neitehr should they be hindered in progressing beyong that if they are capable. It should be noted that facility in algebraic manipulation and elementary calculus is a prerequisite for properly understanding physics and other sciences, and I see no reason why our children should be deprived of such an understanding when they are fully capable of achieving it.

    I am not sure that this actually answers your question, but I hope it explains to some extent where I am coming from.

    Best wishes,

    Sabbir.

    P.S. Thanks for the links, though I am looking more specifically for the bound sets of papers that used to be widely available.

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    Posted by: Betamale 03/01/2011 at 05:38
    Joined on 31/07/2010
    Posts 519

    Sabbir

    Nice post. Could you pen that please and send it to those in power?

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    Posted by: intuitionist1 03/01/2011 at 08:53
    Joined on 10/11/2008
    Posts 44

    Hi,

    Given that "those in power" have been trying their very best to destroy the education system in this country over at least the past 30 years and show no signs of letting up, they would be the last people I would recommend going to or asking for help. I think we need to take things into our own hands and that our efforts are probably better focused towards empowering others to do so.

    As far as mathematics goes, I intend to prepare my own children for Cambridge Pre-U Maths and Further Maths and the STEP papers and NOTHING ELSE and would recommend others with any kind of genuine mathematical aspiration for their children/students to do so. I do not attach any value whatsoever to today's GCSEs and "modular" A levels (or even SATs for that matter) and do not particularly want my children wasting any time revising for them (they are welcome to take them 'in passing' if they so wish).

    Best wishes,

    Sabbir.

    P.S. BTW, these are some of the best of the textbooks that I personally have come across, and currently use:

    1930s-1970s (traditional, 'golden age'):
    "General Arithmetic for Schools" - Durell (1936)
    "Elementary Algebra" Parts I & II - Durell, Palmer & Wright (1920)
    or "School Certificate Algebra" Parts I & II - Durell (1937)
    "Advanced Algebra" Vols I-III - Durell & Robson (1932-7)
    "A New Geometry for Schools" Stages A & B - Durell (1939)
    and/or "Simplified Geometry" - Durell & Tuckey (1933)
    "Elementary Trigonometry" Parts I-III - Durell & Wright (1927)
    "Advanced Trigonometry" - Durell & Robson (1930)
    "Elementary Calculus" Vols I-II - Durell & Robson (1934)
    "Higher Certificate Calculus" Vol I-II - Durell & Robson (1934)
    "A School Mechanics" Parts I-III - Durell (1925)
    [See also, "Certificate Mathematics" Vols I-IV - Durell (1971) (since 1950s)]

    1980s (traditional, straightforward):
    "ST(P) Mathematics", Books 1,2,3A,4A,5A - Bostock et al (1984)
    "Extension Mathematics" Alpha, Beta & Gamma - Gardiner (2007) - for extension work
    "Additional Mathematics" - Forman (1984)
    "Mathematics - The Core Course for A Level" - Bostock & Chandler (1981)
    "Further Pure Mathematics" - Bostock & Chandler (1982)
    "Mathematics Mechanics and Probability" - Bostock & Chandler (1984)
    "Further Mechanics and Probability" - Bostock & Chandler (1985)
    "Pure Mathematics" Vols 1&2 - Backhouse et al (1985)  (actually since 1957 - preferable to Bostock & Chandler)
    "New Tertiary Mathematics" Vols 1&2 - Plumpton & Macilwane (1980) (more challenging)
    (There may be Pre-U or IB textbooks which are also good that I am not familiar with)

    [For primary school arithmetic, I would recommend for example the Bond's No Nonsense series, the Khan Academy website at www.khanacademy.org, the 1971 metric edition of Lovell & Smith's "Two Grade Arithmetic" Vols I-IV for drills and exercises, as well as the set of KS2 Mental Arithmetic books by Schofield and Sims].

    University-level applied mathematics (very useful!):
    "Mathematical Methods for Physicists" - Arfken
    "Mathematical Methods for Physics and Engineering" - Bence, Riley & Hobson

    You will probably have noticed above the unashamed bias towards textbooks by Clement Vasovar Durell, who was the Senior Mathematical Master at Winchester College at the time.  (Note that many of the standard textbooks from the 1920s and earlier seem to be available on the internet, but they are just a little too old-fashioned to be useful other than for historical curiosity. Besides, the expected rote memorisation of much of Euclid's Elements was, I feel, a little too oppressive)
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    Posted by: Nazard 03/01/2011 at 11:19
    Joined on 15/03/2009
    Posts 724

    intuitionist1:
    For example, I recall thinking, back in 1986-88 when I was revising for my A levels, how simple the questions in the exams that I took then had become compared with questions that had appeared in past papers

    This is a great post.

    Betamale:
    A member here has some on his site

    I have spent a couple of happy hours looking at these.  I hadn't seen these past papers before - they are very interesting.

    I have written a huge post - which I'm going to need to carve up.  This one is my thoughts about the O-level papers.

    In the O-level paper from 1940:

    (i) I love the way the very old papers listed the senior examiners (and their names - the idea that the good mathematicians could be marked by "Prof WM Smart", while the less good ones could be overseen by "JW Withrington, Esq" is great).

    (ii)  No diagrams in the very old papers - you had to suss out what diag to draw.

    (iii)  The use of the word "prove" ...

    (iv)  Non-routine problems.  Eg:  A garrison of 1075 men could exist on full rations for 30 days. After 16 days on full rations the garrison is augmented by 129 men and, at the same time, the stock of the existing provisions is increased by 60% by means of parachute supplies. How much longer can the augmented garrison hold out on half rations?  

    1968 O-level paper:

    (v)  This is part of the rubric at the start of the paper:  Credit will be given for the orderly presentation of material; candidates who neglect this essential will be penalised.  Sounds a bit "QWC" to me ...

    (vi)  "Functional Skills" questions, eg: The average yearly rainfall for a certain area is 24 in. During a heavy storm in that area the rainfall was 0.6 in. Calculate the percentage of the yearly rainfall which fell during the storm.  Taking 1 cu ft of water as 6.23 gallons and the weight of 1 gallon of water as 10 lb, calculate also, giving your answer to the nearest ton, the weight of water which fell during the storm on a school's playing fields of area 18 acres. [1 acre = 4,840 sq yd.]

    Then you get to June 1988 (I thought this was the first year of GCSE - maybe the O-level exam overlapped with GCSE for the first sitting?):

    (vii)  June 1988 paper 2, Q1:  A circular fish pond has a radius of 27 m. A sector, of angle 75degrees, is used for fish breeding.   Calculate, to one decimal place,
    (a) the area of the pond,
    (b) the area of the breeding sector.
    [The area of a circle is .... ]
      So here they tell you the formula for the area of a circle!  But they still don't draw a diagram, expecting the candidate to know the meaning of "sector".  Where would this question appear on a GCSE paper?  It would be fairly near the end.  Here it is question 1!

    (viii)  Matrices appear, but then so does some very straight-forward probability.  There is also some "proper" function notation.

    Overall thoughts: 

    • "Dumbing down" is real but not a new phenomenon
    • "Functional Maths" is not new
    • Marks for "Quality of Written Communication" are not new
    • O-levels were clearly and explicitly aimed at the elite - many pupils would score zero on them
    • There was a significant focus on demonstrating the use of mathematical tools, such as being able to manipulate algebra, use function notation, draw diagrams from information given as text.
    • Much of the content was more challenging than we have today and a vital part of a mathematician's toolkit (matrices, Newton's Laws, calculus)
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    Posted by: Nazard 03/01/2011 at 11:26
    Joined on 15/03/2009
    Posts 724

    Nazard:
    I have written a huge post - which I'm going to need to carve up. 

    Here is the second part - about GCSE papers.

    I used to reply to the people who claimed maths had been dumbed down by pointing out that the syllabus (as it was called once) had changed beyond recognition so it was impossible to compare O-level exams with GCSE.  I was wrong.  Here are the possible positive points about recent GCSE exams:

    • There are different topics involved.  Yes - to a degree.  Probability didn't used to be in O-level, nor statistics.  It feels to me as if the former should be there but the latter probably isn't "mathy" enough to qualify.  They are easier than the things that have been dropped (matrices, etc)
    • Attainment Target 1 didn't used to exist.  Yes, but if you look at GCSE papers since coursework has been removed, there is not a significant amount of U&A in the exams, and certainly no more than there was in the O-level papers.  A question that now might state: "Work out the length of the missing side in the diagram" that shows a right-angled triangle would be claimed by the examiners to be a "choose what maths is appropriate, because Pythagoras is not mentioned".  This sort of question appeared in the O-level papers, but in words rather than with a diagram.
    • We do lots of investigations in lessons now, though.  Yes - but in many classrooms they don't because this sort of thing isn't tested on an exam so lots of teachers believe there is no point.  The way early GCSE exams contained 'mini-coursework' extended questions meant this was perceived to be important and was taught.  Nowadays, the exam questions are so easy there isn't much of a perceived need to teach this.
    • Ah, but Functional Skills are much harder.  Yep - but they were in the 1968 O-level paper.
    • But what about the pupils who won't be able to access the questions - this is all very elitist.  No it's not.  There are several issues here.  First of all, everyone needs acess to an exam on which they can show what they can do.  For some, this will be an Entry Level Qualification.  For others this will be GCSE Foundation Tier.  Some might need FSMQ, etc.  Secondly, no-one should have an artificial 'ceiling' imposed, so there should be opportunity for pupils to sit the exam that is most appropriate for them, with a decision taken fairly close to the exam as to which this will be (which will mean a significant overlap in material between the different options).  Thirdly - GCSE results today are in no way comparable with O-level results of the past.
    • But Edexcel tell us that their exam papers are just "accessible" and that they "allow pupils to show what they can do".  Yep - by providing diagrams, mickey-mouse questions and very little challenge.
    • Back to the elitism thing: your able pupils can just take FSMQ or A-level modules and the rest can do GCSE, so what is the problem?  Well - there used to be this system where there was a reasonably rigorous exam for able pupils (it was called "O-level") and an exam that could be compared to it but which was for less able pupils (the CSE).  Maybe the problem was that these weren't implemented well, but if pupils could take the one that is appropriate for them (*) then this strikes me as being a much better system than the current one.  (* so it isn't the cae that Grammar schools do O-levels and everywhere else does CSE)
    • Why can't they just do FSMQ, though?  They can.  But they still have to take GCSE papers.  And the current grades are absolutely not comparable with the ones that O-levels used to produce.  Surely that is not helpful?
    • Exam Boards (as they used to be known) provide lots more support than they used to.  Yes they do - and some of this is good.  Being able to access individual marks that pupils got on particular questions is brilliant.  Having textbooks written by the examiners isn't (for reasons I have pontificated about in other threads in the past).  Having training provided by the examiners isn't right.  Having a free market isn't. Back in the olden days the universities set the exams and didn't really care how many people took them, or what the results were like.  Nowadays the exam boards do care and need to make sure they are pushing results up slightly higher than other exam boards so the paying public will continue to buy their product rather than that of the competitors.

     

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    Posted by: DM 03/01/2011 at 14:27
    Joined on 12/05/2003
    Posts 5,447

    I still don't really understand your agenda Sabbir but here are the STEP papers you were after:

    1987 - 1999 http://bit.ly/9xCtXO

    1998 to 2010 http://www.admissionstests.cambridgeassessment.org.uk/adt/step/Test+Preparation

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    Posted by: Batsheep 03/01/2011 at 17:14
    Joined on 17/06/2008
    Posts 395

    Somewhere I have my grandmother's 1923 school certificate papers in arithmetic, algebra and geometry.  I'll see if I can find them, scan them in and get them to you.  I tried some of the algenbra out on an A level Maths class once and there hadn't got a clue.  (The geometry and arithmetic were so far off their knowledge as not to be worth trying)

    I also have her RE paper with optional questions in Greek and her French where students were meant to translate chunks of Moliere.  No dumbing down since then!

     

    It might take me a couple of days to get around to this, but I'll contact you and post on here when I've got them.

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