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davidmu:

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.

They should be able to do it if they have studied roots of polynomials in AS further maths david.  The allocation of topics to units has changed, that's all.

intuitionist1:

An AS level student today may be able to plot x/(1+x)^2 easily on his calculator by typing in the equation and pressing a button, but would probably be clueless if asked to sketch the curve. On the other hand, a 1957 AO level student would be able to sketch the curve complete with maxima and minima and asymptotes.

They'd have a job seeing as it doesn't have a minimum point.

becka frequenty assumes that every school has mymaths too and I am sure this is not the case.

intuitionist1:

I am not sure what the point you are trying to make here is. An AS level student today may be able to plot x/(1+x)^2 easily on his calculator by typing in the equation and pressing a button, but would probably be clueless if asked to sketch the curve. On the other hand, a 1957 AO level student would be able to sketch the curve complete with maxima and minima and asymptotes. As you say, they would find learning to use the calculator just as simple as it students today if calculators had been available at the time.

But these are mechanical operations. I would much rather have a student who can plot a curve on a calculator and actually understand what it all means, rather than have a student who can sketch a curve, work out maxima and minima without any real understanding of the mathematics behind it. Naturally a combination of both worlds would be ideal

intuitionist1:

I don't really agree with this. The reason the question is relatively difficult is because it requires thinking several steps ahead, not because it involves sums and products of roots. I don't think today's students are currently as well prepared in that respect, which is a shame.

I think you are wrong. I could quite happily give such a question to my students, explain the definitions required, and leave them to it. It is not a particularly difficult question.

Hi Rebecca, 

weebecka:

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.

I am not sure I understand. Are you referring to mathematical intuitionism (introduced by Brouwer and formalised Heyting et al) and constructive mathematics? Is this related to your concept of "high road" teaching in some way that I am missing?

weebecka:

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.

Okay.

weebecka:

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. 

But surely getting students to work where they think for themselves is one way of increasing the difficulty level - I don't see why these are mutually exclusive? I agree that getting stuck (and then unstuck) and dealing with confusion is a valuable part of the learning process. So the difference between "low road" and "high road" teaching is the difference between straightforward applications and non-trivial applications (i.e. 'extension' work) of a topic?

weebecka:

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.

I agree that computers can be used to track student progress and can aid in reducing time spent in lessons on practice drills etc, but I don't think that sitting in front of a computer and solving problems is better than solving problems on pencil and paper without a computer in sight. Quite the opposite in fact. I agree that it is useful to have students practice outside of the classroom - they can be given exercises and drills to do for homework for example, but see no particular need to introduce computers for this purpose when there are textbooks that can do the same job. I do not agree that learning the core ideas should be left to a computer, not even mymaths (which one of my sons sometimes uses, but which I find to be really dumbed down, limited by its very nature of being a fixed piece of code, and generally irritating). The reason like Khan Academy is because Sal Khan does a pretty good job of teaching in his videos, and the user interface is completely stripped down to its basics without any graphical gimmicks.

weebecka:

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).

I think we were probably talking about completely different things here.

weebecka:

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.

Okay, I've taken a brief look at this, and I think I see what they are trying to achieve, but I also think that it is a misguided approach despite their good intentions, and is more likely to confuse students with regards to basic concepts when they should really be being given clear and well-defined guidance. Well-designed questions will well-defined (not open-ended) answers will help students pick up the same skills without risking them wandering off into never-never land. Take a look at any of the textbooks I listed earlier by Durell and get your students to work through the exercises and I guarantee that you will get better results than having them mess around with the open-ended expeimental projects on this website.

weebecka:

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?

I think that the converse is actually true - that if the subject is well taught (by a human being), students have no need for ICT. I do agree that computers can play a roll in providing practice questions and assisting the teacher to track progress of students, but I simply do not like the idea of students doing mathematics in front of a computer (or indeed any other subject, except perhaps computer programming - I _would_ encourage some level of programming to be taught at an early stage to those who are capable of it).

weebecka:

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.

Okay, but I doubt that it is likely to change my opinion on the matter.

Best wishes,

Sabbir.