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intuitionist1:

Karvol:

 Jeez Sabbir, you have a lot of time on your hands.

I used to work in the financial sector, until I decided no amount of pay was worth the hours and I liked long, long holidays. Saturday used to be one of my busiest days.

Have the hours been reduced in the last 15 years or so?

As far as I know, I have the same amount of time as every else - though I do like to use it well. We don't live for very long and there are a lot of things I would like to do while I still can (most of these tend to revolve around education in various ways mainly because of the children who are growing up fast, though I do also like to keep up with research in theoretical physics, statistics, and various coding projects, and tend to toggle between these periodically).

Banking hours are 8am to 6pm weekdays (weekends free) where I am currently working. Others do work much longer hours than this - depends upon the role. (Were you in trading?) At least the hours at the bank are regular, which cannot be said for consulting generally. I used to be a equit derivatives quant, which was hard work and eventually got a bit tedious (a bit like accountancy for theoretical physicists), but I am currently working on the strategy side of things on behalf of senior management, which involves all kinds of work from statistical analysis, coding and system development through to higher level business modelling and strategic planning. Certainly keeps the brain active and the creative juices flowing and ties in well with things I have done in the past as well as other projects I am currently working on as a hobby.

Best wishes,

Sabbir.

Trading. VERY briefly. I threw the towel in when I was asked to come in at about 6 am on a daily basis.

DM:

Thanks for that Sabbir.   Paper 6 is considerably more difficult.   I particularly the geometric application of the palindromic quartic (number 8).  

This still shows STEP in a good light as a decent examination of able students.

Besides the obvious increase in difficulty compared with today's papers, I think there is a huge qualitative difference in the way questions are asked and what is expected of students in terms of cognitive ability.

I also managed to get hold of a University of Cambridge GCE A & S level book for 1951, which contains papers for all subjects (including "Handicraft" :), and it is a real gem. The format of the papers is much more familiar to me - similar to the exams I took back in 1988. Browsing through the questions, it looks like they are of a similar level to those I remember from the late 1960s, so I guess that standards were generally maintained up to around 1970 after which the complaints began. I note that there are only two papers (first seems to cover algebra and calculus and the second covers geometry, applied maths and statistics), and one scholarship paper at A level compared with the 5 subject papers from 1918, though the subject is now split into two levels of difficulty - Mathematics and Further Mathematics. It looks like the 1918 distinction papers were the equivalent of the Further Maths A and S papers of today, though my general impression is that the 1918 ones were harder.

I am going to head off home soon, so I am not going to type out any more questions. However, these are the headers of the papers for comparison with the present day's miserable offerings:

==================

University of Cambridge Local Examinations Syndicate

General Certificate of Education

Mathematics, Advanced Level, Paper I (Three hours)

Full marks can be obtained by complete answers to about ten questions, but a pass mark can be obtained by good answers to about four questions or their equivalent.

Write on one side of the paper only.

Mathematical tables and squared paper may be obtained from the Supervisor.

1. The pth term of a progression is P, the qth term is Q, the rth term is R. Show that, if the progression is arithmetical,

P(q-r) + Q(r-p) + R(p-q) = 0,

and that, if it is geometrical,

(q - r) log P + (r - p) log Q + (p - q) log R = 0.

...etc. (12 questions on paper)

=====================

Mathematics

Advanced Level

Paper II (Three hours)

Answer the whole of Section A and four questions from Section B which include any of the questions on statistics.

Full marks can be obtained... (as for Paper I)

=====================

Mathematics

Scholarship Paper (Paper III) (Three hours)

Full marks can be obtained by complete answers to about ten questions.

1. Prove that

2 cos x/2 (cos x - cos 2x) = cos x/2 - cos 5x/2

Solve the equation

cos x - cos 2x = 1/2

giving the values of x which lie in the range 0 <= x <= 2 pi

etc...

===============

Finally,

Furtyher Mathematics

Scholarship Paper (Paper III) (Three hours)

[As above]

1. (i) Prove that the sum of the first n terms of the series

sin x + 2 sin 2x + 3 sin 3x + ...

is

((n+1) sin nx - n sin (n+1)x)/(2 (1 - cosx))

(ii) Find the sum of the first n terms of the series

1 - 5x + 9x^2 - 13x^3 + ...

and write down the sum to infinity if |x| < 1.

etc...(12 questions).

======================

Okay, that's enough for today. Have a pleasant weekend!

Best wishes,

Sabbir

intuitionist1:

I will see what I can do about making some of these papers available. I don't realistically think I will be able to spend the time to sit in the library Latex-ing them all (today is very much a one off as my car is currently being serviced!), so sorry for that. I hope that the above is sufficient to whet your appetite for the time being.

Best wishes,

Sabbir.

Sabbir. This would be awesome if you could get someone to either do it or locate its...sounds like a nice job for a budding work experience student.

Thanks for the ones you have posted though

weebecka:

Apparently not curls.  I was struggling for credibility early on, as you remember, so I asked them and this was my reply.

Early on? do you believe thats changed?

Don't worry about my reputation darling, its just fine. 

Why do you respond in this manner to so many questions? Seems odd. Its a cross between being patronising and upper middle class interaction...or maybe its just the former.

Your waffle on 'High Road' teaching is yet another example of garbage produced by someone who doesnt teach in a full time job and preaches the method to others.

Its flaws are simply hidden by fancy terminology which most won't question.

Whats next on the menu? Rocky Road?

Hi intuitionist1, Yes it would be great if you could make those papers avaiable.  Now, where do we begin with the big post?

Okay, well firstly, by your name, I thought you had more understanding into this already than you do.  Why the name, intuitionist, if you do not understand intuitionism?  Oh well, never mind.

intuitionist1:

I think that people naturally make connections with past experience, and often across different subjects

Yes they do.  But teaching strategies which plan for this greatly enhance the rate at which it takes place.

intuitionist1:

It seems to me that introducing students to a wide range of different problems in a graded fashion from simple to complex would naturally help them to build up such a mental network of interconnections.

Absolutely.  Students should be rigorously taught an axiomatic curriculum.  This is the low road teaching.

intuitionist1:

With experience, students will gradually build up expertise in recognising patterns and connections even between topics which may naively appear to be widely different. This allows them to draw upon a wider range of problem-solving techniques and take a more holistic or lateral approach when straightforward attempts fail. As they gain in experience, they will become more efficient in finding the optimal approach to solving each new problem, and also becoming increasingly expert in reducing real world problems in diverse fields into a well-defined mathematical representation which they can proceed to tackle with the battery of tools that they have developed.

If you simply build examples in order of difficulty, you do not achieve these aims.  To achieve them you need to get students to work in where they have to think for themselves.  They have to deal with confusion, they have to deal with being stuck.  This is the high road teaching. 

We should do both.  Interactive online technology now helps us subordinate the former more as it can deliver and personally track 'low road teaching' more effectively than before, creating more classroom time for 'high road teaching'.  Technology can now take the strain of ensuring studenst practice and become fluent in the core techniques (the low road).  When you were supporting students, did they show you mymath (mymaths.co.uk - most students have access).  High road teaching rapidly and effectively identifies gaps which are often rapidly and efficiently filled in context.  If not, the student can do the relevant mymaths lesson, or at higher levels they can watch the soon to be relaunched Khanacademy.

intuitionist1:

Developing such intuition however takes a long time and requires a lot of hard work, and it seems to me to be more appropriate for PhD students than today's A level students.

There I disagree absolutely.  It should be part of the education of all mathematics students at all levels.  It helps if students enjoy and are confident with mathematics and for must students it is the high road teaching which gives this enjoyment through creativity and this confidence through flexility (although often there is an uncomfortable transition phase if students have had low road teaching only for many years).

intuitionist1:

What are the students expected to learn here?

The best way to understand this is to try it.  That exercise comes from the list of rich tasks at A-level which you can find at www.risps.co.uk  From that site you will also find a link to rich statistics tasks at a-level (making statistics vital).  So you should be able to find a task which will suit the topic you are teaching at present.

intuitionist1:

my opinion of ICT is, in general, that it should be avoided if at all possible in a teaching context

Correctly used ICT has many benefits for mathematics education.  But rather than listing many I want to focus on just one today (you can always challenge me for more later).  Ask the studenst you teach to show you mymaths Sabbir.  Look at the lesson on composite functions in C3 which I would set as a homework in conjction with the rich class task I suggested earlier.  Understand that their teacher can view from any computer with interent their precise progress - how many times they've tried the exercise, which questions they've got right and wrong and so on.  They can see class summaries at any time.  The teaching in the interactive lessons is very good - have a look at a few more of your choice.

Are you absolutely sure this is not beneficial in the way I have described Sabbir?

Give your reply, I also suggest you have a look at the thread on 'The concept of a function' started by Dag Rune in the LinkedIn group.

GCE Ordinary level, Summer 1957. paper d, London

Q1, If alpha and beta are roots of the equation x^2 +px + q = 0 and beta and gamma are the roots of x^2 + rx +s = 0, prove that

(a) alpha + p = r + gamma,

(b) q * gamma = s * alpha

(c) alpha^2 - gamma^2 = (p + r)(p - r) -2q + 2s.

I think few of today's students would manage this or indeed any of the remaining questions from this paper.