Tag Archives: mathematics

Drilling for success – a reflection on the GCSE results

I’ve been reading the reports of the summer GCSE exams – not the press reports on the issues surrounding GCSE English, but the reports that matter to me, teaching GCSE maths.  Thanks to technological innovations, and a certain forward-looking exam board, we get two types.  The first is the overall report of the exam itself, with grade boundaries, pass rates, and the chief examiner’s comments on the papers and the individual questions.  The second is the local analysis that provides information on the institution, classes, and individuals down to the level of each question. So for example I know that we performed better than the national average, we exceeded the average on almost every topic, and that candidate W got question one wrong.  And what strikes me about the reports is not the self congratulation of being better than average, but topics on which nationally so many learners perform badly, and the examiners comments on these issues.  And I find it rather disconcerting.

And equally worrying is the fact that the chief examiner appears to be using ‘cut and paste’ over successive years. The following appeared in 2009 and again in 2011 in identical form: ‘All candidates should have a calculator for the calculator paper. Examiners become very concerned when it is clear that candidates are desperately trying to solve complex problems by hand, suggesting a calculator is not available.’ Do we learn nothing, us teachers?

A few years ago coursework was abolished for GCSE mathematics, with the comment from the examiner that we could ‘now concentrate on teaching’.  I was worried then that someone within the examination system was dismissing the benefit of investigative learning.  So many teachers see our job as one of teaching, training and drilling, with little focus on learning.  And things haven’t improved much – a recent comment on a badly answered question demanded that ‘This needs to be taught more rigorously than before.’  And when a question has been answered well, credit is given to the teacher: ‘Many candidates had clearly been drilled into the correct form of words, and for them this was an easy mark.’

But it is clear some topics have not been sufficiently drilled. In June 2011 ‘The difference between perimeter, area and volume remains a mystery to many candidates.’ And asked to draw a rectangle of a given area in 2012, incorrect answers showed ‘…confusion between area and perimeter. A very small minority drew a triangle instead.’ There remains then an issue with simple shapes: ‘Two third of candidates could not write down the mathematical name for the quadrilateral.’ It was actually a trapezium, and a drawing of one is on the inside cover of the paper, with the formula for finding its area. Other technical terms confounded the candidates, many of whom could not differentiate between parallel and perpendicular.  And three dimensional shapes fare worse. ‘Not many grasped what the question was asking. It was clear that many candidates struggled to visualise what shape would need to be added to make a cube.’

Shape and space questions may throw up some serious misconceptions, but other topics are equally susceptible to a lack of understanding of the technical terms.  In  2012  ‘Many confused median with mean and mode.’ And I rather like the uncertainty about certainty – I’m quite happy to believe the sun will come up tomorrow, but apparently we need to make this more clear to the learners: ‘Candidates should be advised about the practical interpretation of likelihood, e.g. that although nothing can be considered to be truly certain, like the sun rising tomorrow, that for all intents and purposes the probability that the sun will rise tomorrow is as close to certainty (i.e. unity) as makes no difference.’

It seems to me that it is an issue of communication, but I’d welcome discussion.  If learners can’t distinguish between words, isn’t it because they have not had sufficient opportunity to use those words in the appropriate context, rather than not having been ‘drilled’?  (And it’s so long since I used the word ‘drill’ I’m not sure how to do it, not in a learning environment.)  What do I do to be better than average?  Sit back and let the learners do the talking – let them make and then name the shapes, let them count the vertices, edges and faces, and report their results.  I set investigations involving area and perimeter, which they share, using our special mathematical language.  I bring in the learners own personal data, which we share and analyse, and I hope to give as an example when we do it again this year.  And we talk about events and likelihood, so that I’m quite confident that my learners appreciate that statistically speaking, the sun will come up tomorrow.

What I don’t do is use classroom time for repetitive examples from books and worksheets, or actually anything that asks the learners to work alone and not communicate with one another.  And on what did we do worse than average?  We dipped below on a handful of questions on manipulating simple algebraic terms.  I’ve hopefully addressed this with some kinaesthetic card matching activities that combine shape and algebra questions, which incidentally was a question on one of our summer papers.  It proved difficult for almost everyone. ‘The last and most challenging question on the paper had only had 4% of correct responses with many not attempting the question at all.’  So I’m learning from my mistakes, as hopefully the learners do, and plan to do better next time, which for many learners is not an opportunity that comes their way.

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Rich Questions

Rich Questions

Many years ago, in the early days of Inset and ‘Baker Days’, when people still knew who Baker was, a colleague gave a demonstration of a simple resource, an apple, and how it could be used to enhance his teaching of modern languages.  I’ve never forgotten it, as the epitome of a crass and thoughtless teacher showing off his knowledge without a care for our bored minds and beleaguered bums.  He spoke without pause, and we probably forgot every word he said.  I’m entirely in favour of simplicity, and love a simple question to start things off, and watch the threads take off in a variety of directions; in fact, the simpler the better.  My group on Friday afternoon are returning learners of assorted ages, nationalities, and existing skills.  I’m confident all of them have a wide range of mathematical knowledge, and my job is to get them back into critical thinking mode before term starts in September.  We’ve covered some algebra and some ‘shape & space’, and lots of incidental number stuff in the first couple of weeks, and this week returned to shape.

Me – ‘Draw something with an area of 18 square centimetres.’

I’d supplied centimetre square paper, rulers and pencils, and the rest was up to them.

After a few moments:

A – ‘I’ve got them all – rectangles 1 x 18, 2 x 9, and 3 x 6 – that’s all there is.’

C- ‘What about 4 x 5? 4 + 5 + 4 + 5 = 18?’

A – ‘No, you are confusing area with perimeter – area is the inside, perimeter is round the outside.’

(A demonstrates to C on C’s drawings the difference between the two concepts.)

Me – ‘Are you sure this is all?’

Another pause…

C – ‘Can we use halves?’

Me – ‘Did I say we couldn’t?’

C – ‘In which case we could have 4 x 4.5, 1.5 x 12…’ and goes on to list numerous other cases. By this time, C has moved entirely from the concrete to the abstract, and has ceased to draw rectangles.

Meanwhile, B is thinking…

B – ‘Can we have a square?’ And answering herself: ‘No, because the side would be the square root of 18, and that isn’t a ‘real number’’. She might not have said real, but something like it, indicating it wasn’t something we could measure with a ruler.  I actually take part for a moment, stop asking questions and open a calculator on the interactive board to show that root 18 is a damned tricky number, if you want to catch hold of it with a calculator.

Me – ‘You might like to come back to that one.’

B- ‘What about triangles? Can we have an equilateral triangle?’

Me – ‘That might require some serious mathematics, but there are lots of other types of triangles that you might look at.’

C then produces a number of right-angled triangles, and demonstrates to B how she has halved assorted rectangles of 36 square centimetres to produce triangles of 18 square centimetres.

Meanwhile, A is ordering her thoughts around the number of variations on the original question.

Me – ‘Are there an infinite number of shapes with that area?’

A – ‘Yes.  I’m thinking of a splodge, and that splodge can be any shape at all that covers 18 squares.’

Me – ‘A splodge – I like that.  Sticking with rectangles for a moment, are there an infinite number of rectangles?’

A – ‘Yes there are.  If I have an area of 18, and a given length, I can divide to find the width.’ We are now into re-arranging formulas, by intuition.

Me – ‘What did we refer to when we were looking at patterns last week, and general terms?’

C – ‘A variable.’

So now I’m inclined to open a spreadsheet on the screen, and A demonstrates confidence and fluency with Excel.  I put in some headings, and she explains the formula for the ‘Area’ column.

A – ‘It is the contents of the cell A2 multiplied by the contents of the cell B2.’ Spoken like a pro, and delivered with more clarity than a good many IT teachers could muster.  We pause whilst A explains the concept of spreadsheets to B and C. She then does all the clever bits, like copying down formula, and creating a new sheet for doing it backwards, with area fixed and length as the variable.  After which A and I get into a technical discussion where she initially sees ‘infinity’ as going on forever, and I’m hoping for her to see the infinite number of cases bounded by the ones she has.  In other words, she can see rectangles stretching forever and getting thinner, and I’m seeing an infinite set of widths, say, between 4cm and 5cm.  But time is pressing.

Back to D, who by now has moved from right-angled triangles to isosceles and scalene triangles, all the while rediscovering her knowledge and naming those shapes.

M- ‘What about the square – is there any way to construct it, without having to measure root 18, and thinking of what you have done to find the triangles?’

D – after a very short reflection: ‘Yes, if you draw a square of 36, then cut off the corners to create… What’s it called?’

A – ‘A rhombus?’

D – ‘Yes, a rhombus,  a square that’s half of the square, another square inside it, tilted…’

Me – ‘Come up and draw it!’

She does, and we have to break off shortly, with the briefest recap of what has ensued, what has been revisited, learnt, expanded upon.  It’s show time, and C and B have a suitable array of rectangles and rectilinear shapes to share.

 

And an hour was far too short. What could we have done with two, and what might we do next term when they start for real:

Do all the shapes have the same perimeter, and if not, which has the least?  Is there a maximum?

Will we find that parallelograms are easy to compute, and are trapeziums just as easy?

Are there any simple regular shapes that have an area of 18 sq cm, apart from a square (which wasn’t that simple at all), or do they all behave like equilateral triangles and need skills we might not learn in a few months?

Will we get to circles, and rearrange that formula that Maggie Thatcher said every pupil should know, to find a circle of 18 sq cm, and has it the smallest perimeter? (Cherry pie delicious, apple pies are too.)

And can we generate the formulas for the ones we’ve found, producing simple elegant statements with symbols and operators?

And could we use those formulas to generate variations, as we rearrange them according to known variables?

I’m not entirely pleased with myself, although I’m pleased that everyone has shown learning, and that both A and D have demonstrated true mathematical thinking and insight. B was not confident that he’d contributed, but the others reassured him that he’d be supported in future.  I think I was indulging, and could have possibly tried to keep things simple at this early stage. And next term I will.  But there again, challenging is good.  And I had fun, and remember, that’s the important bit, I’m the teacher!