Geoff Tennant - Promoting access to mathematics for all
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4/6/17: thinking whilst checking A level answers

Very pleased to have been checking the answers this last week to exercises for a further mathematics (pure) A level textbook due to be published shortly.  Many thanks to RT for making the connection, most grateful to you!

The work raised a whole number of issues, some new to me, some old, and some adaptations from other contexts.  So firstly, a question which I have grappled with from time to time over the years: why does mathematics need 2 A levels whilst every other subject has just the one?  One problem here is that this is one of those questions which it's difficult to address away from our own experiences.  So just briefly: I myself did further mathematics A level (als physics and, unusually at the time, English) before going on to a matheamtics degree, and then taught further mathematics A level in my first 5 years but not since as my interests in mathematics have become increasingly in the learning and teaching of the subject, particularly at upper primary / lower secondary level, rather than myself doing hard sums.  From a teachers' point of view, if this is what you're used to teaching then you're likely to quite enjoy doing it, from the perspective of Universities doing mathematics and subjects drawing heavily on mathematics, eg. engineering, then having students joining with a solid grounding in the subject is clearly desirable.  Not to mention the people who write textbooks, examination questions, etc. etc. and otherwise have a stake in the subject, up to and including whole careers.

In my experience discussing the questions as to the need for a second mathematics A level, most reasons given boil down to, "This is how it was done previously so this is how it must continue to be done."  So an example of this is, "Students need it when they go on to University."  Yes, but if writers of the syllabus for numerate degree subjects knew that they couldn't assume material currently in further maths A level, they would have to teach it themselves, wouldn't they?  Conversely, if historically there had been two geography A levels, physical and human let's say, ,then exactly this argument could be used to support this continuation.  More generally, this really is not a good argument, as I argued in Uganda last year - see this blog post - this argument can be used to justify the teaching of logarithms as a calculation tool long after calculators have made this redundant.  (I said this in the context of current reforms dropping this topic, I'm pleased to say.)

To my mind, the best argument I've heard is related to the different ways that different subjects can deal with their highest attainers.  So, for example, a strong student in English literature studying one Shakespeare play can read other plays written at the same time, find other accounts of the historical background, read books of literary criticism, etc.  In mathematics, if you're understanding a topic and routinely getting questions right, then that is as far as you can go before learning new subject material.

Am I convinced?  No, not totally.  Had the idea for further mathematics come after AS levels - ie post 16 qualifications with half the content and the same rigour as A levels - a solution would have been to have further mathematics AS rather than a full A level.  It is already the case that, relative to other countries, post 16 students in the UK specialise to a very high degree, the 2nd maths A level accentuates that, putting 16 year olds in the position of making decisions which seem to me to be premature.

So, to the work I was doing last week.  First of all, the idea of getting somebody involved at proof reading stage, not previously involved with the writing of the textbook, is superb.  Doing the job properly means actually doing the questions, I have great sheaves of A4 paper to prove this is what I did should anybody wish to check up on me!  It is the nature of mathematics that everyone will make mistakes while working, whilst it is possible that two people working totally independently will make exactly the same mistake it is extremely improbable and a really good check before going to printing.  There have, over the years, been problems with mistakes getting into examination questions, most recently over whether Tybalt in 'Romeo and Juliet' was a Montague or a Capulet, it is exactly this kind of check which is needed to minimise the possibility of this happening.

(Parenthetical thought: in the aftermath of the problem with the English A level question, a representative of the examination board said something like, "We will ensure that no candidate is disadvantaged by this mistake."  Well, forgive me, but this is clearly total nonsense, isn't it?  It is reasonable to suppose that some candidates were not intending to answer questions on that play, others would go to it first.  Some would spot the mistake and have the confidence to write their essays according to a corrected version, others would be completely thrown, with their entire knowledge of the play undermined.  How possibly can equity in this situation be ensured?  End of parenthetical thought.)

So my job was to go through the textbook, doing the questions, and then compiling a list of mistakes and sending them to the publisher.  The standard in the textbook was pretty high, which of course is good. But this meant that I worked for hours, checked the answers, all of which matched, so from the point of view of reporting back I had nothing to show for the time.  I felt I wanted to regress 40 years, "Look, miss, I am doing my homework properly, really!"  So to make myself feel a bit better I stuck in an occasional comment when I thought the questions were a bit tough.

But there is another issue here which I have thought a bit about over the years concerning mathematics at earlier stages, it was interesting to think about it at further mathematics A level standard. The analysis of mistakes made is a fascinating area, one crucial distinction to make is between consistent and inconsistent errors (also called careless errors or slips).  So, if a child makes a mistake, a common teaching tactic is to say, "Can you read that out to me?"  Or, "Are you sure that is correct?"  If the child takes another look and can immediately see the error, then this is a careless error.  If they insist that they are correct, then this is a consistent error relating to a misconception somewhere.  It can easily be seen that how the conversation continues from this point will be very different according to how the child responds.

(Parenthetical thought: as an initial teacher trainer, I addressed the point with my students that a problem here is that children come to learn that when a teacher asks, "Are you sure that is correct?"  they mean that there's a mistake. So I encouraged my students to ask these kinds of questions then what the child had written was, in fact, correct.  Be warned, children may well find this confusing at first, but well worth persevering, the issue of confidence levels alongside being right or wrong is an interesting one but beyond this blog post, alas.)

A related issue here is that, certainly up to GCSE (equivalent to O level) standard, it is considered good practice to test one skill in any one question.  If, for example, you want to test understanding of equivalence and cancelling of fractions, then you write a question in which the arithmetic required is straightforward, the more difficult the arithmetic, the greater the chance of a careless error, the more you are obscuring whether an incorrect answer relates to a conceptual misunderstanding or a slip.  Whilst doing the exercises I found myself having to do some quite lengthy calculations, where my initial answer differed from that given I was quite often able to go back over my working and find a mistake somewhere.  Sometimes a simple answer arose from complex calculations, which often (not always!) means that one has completed the work correctly, but sometimes the correct answer really was not very beautiful.  From the point of view of A level students working through this, I'm wondering if constant wrong answers arising from slips can be a bit dispiriting - but against that I am aware that the level at which these students are working is high and should not be expected to be easy.

Not entirely sure where I'm heading with this, although I am left wondering whether principles of good practice at primary and lower secondary level could be better applied to further mathematics A level standard.  Happy to discuss this point further with anybody interested!

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