## What’s missing from Maths GCSE?

Calculus. A word which excites me, probably more than most people. I’m amazed by it. It’s conceptual brilliance, it’s abstract nature and it’s real life application. Not just that though, but it’s history too.

From the earliest visualizations of Archimedes and Diophantus, through the work of Fermat, to the amazing work the student Newton did to really bring forward the calculus we know today. Leibniz and the plagiarism battle. Then the rest, Newtons further work: the product, quotient and chain rules, integration by parts, multivariable calculus. All of it.

Calculus sits alongside complex analysis as my favourite areas of maths. With that in mind, it is probably no surprise that I lost a few hours on Friday to planning a course of lessons entitled “Introduction to calculus” for my year 12 class. (I will share when finished!) In fact, I was so excited by it I ended up dreaming about calculus. (Not for the first time…. I remember while in my second year at uni cramming for an exam on a tensors module and dreaming I was floating through space surrounded by integral signs….)

Then on Saturday I perused the new proposals for A-Levels (Not maths, we’re getting our own review later). The science A-levels have a decent amount of maths in them, it was good to see. However, I noticed there was mention of rates of change and graphs and calculating them by drawing tangents and using them to approximate the gradient. This reminded me of something that I’ve long thought: “Calculus should be on the GCSE specification”. Calculus is the mathematics if change. It can tell us exactly what the gradient of the curve is at a given point and it cab tell us exactly the area. It doesn’t get used in science A-levels as it doesn’t appear until maths A-Level and we need these to stand alone. So why not solve it and teach it to those in compulsory education? Basic calculus is no harder to grasp than circle theorems. Why should it be kept solely for a level? Why shouldn’t it be the spark that inspires more to study maths A-Level? It was on the old O-level when my parents were at school, it didn’t do their generation any harm. In fact their generation produced some amazing mathematicians such as Andrew Wiles.

When I heard about the coalition government’s review of maths GCSE I had high hopes for it. I hoped calculus would make the cut. I was disappointed.

There are other things missing too. Take Standard Deviation, for example. It is the building block for most of the statistical analysis that goes on in psychology, biology and other areas, but is not taught as standard to those in compulsory maths educational.

I would like to see group theory on the GCSE, set theory too. Perhaps a little bit of matrices. Matrices and critical path analysis were then when I did it, and I’ll never know why they went. They are fun and hold far more meaning than some things we teach today, such as trial and improvement!

Are there any topics you’d like to see in there? Or any you’d like to cut? I’d live to hear them and your reasoning.

Standard deviation should definitely be on the GCSE syllabus – it’s pretty straight forward to get the basics, and it’s something that does have real, practical applications in a lot of situations. I’m forever plugging stdev(range) into Excel to flag up significant outliers – no, it’s not the most robust analysis but it’s good as a “back of the fag packet” method of quick calculation.

Some very basic introductions to calculus might be worthwhile at GCSE, but to be honest I felt they were better left to A level other than as extension work for the A* candidates. Complex numbers, on the other hand, would be a good addition to the GCSE syllabus, because they help to explain all those errors that calculators have been giving you all along.

I agree entirely about SD. As for complex numbers, I’d not considered it, but now you have mentioned it I am wishing they were there!

Hi

Great piece and something that as a HoD for Science makes me think that some joined up curriculum planning is in need to genuinely move education on.

We (Science) would love the introduction of some calculus into GCSE as it makes any scientific process that involves rates of change far more generic to teach – and thus give transferable skills. Any students taking Science post 16 would be enhanced by being able to calculate gradients at particular places on a known curve – via calculus rather than tangetn drawing.

You mention axes drawing in another post, and this has long been a bug bear – with students taking longer to puzzle out the scales than to draw the points. Even then, they draw them wrong. We’ve focused on the mechanics of plotting points rather then the purpose of plotting them for far too long.

And as for statistics – imagine in an A-level biology class having to teach Students-T distributions to a class of learners who achieved a C grade in maths by taking the foundation paper. Statistical literacy is as important as numeracy in today’s world — 11 out of 10 cats say so.

I’d love to write a new GCSE that takes Maths, Science, Statistics and Engineering and join them into a new qual, GCSE Experimentation!!!

Good call

Glen

http://www.glengilchrist.co.uk

Hey Glen, thanks for the comments and I couldn’t agree more. There is not enough joined up thinking going on in the DoE! This curriculum review was an amazing opportunity which has been missed.

Before retirement, I was cheerfully teaching complex numbers to bright 10 and 11yo’s (at a state inner-city-except-we-didn’t-have-gangs Junior School. At GCSE, I’d love so much more stats – SD – outliers as Stevie said (look – I wish the stats in the Spirit level were correct as much as anyone – but I’d rather students knew they weren’t) Much more sophisticated probability – again, cos that’s a skill that would make for so much more an educated population. I’d love a return to formal geometry, as that used to be the standard introduction to logic and reasoning.

And not GCSE per say – but Sets – what happened to them – in Y1 you separate objects into sets, say by colour – and that’s it. If I hear one more teacher use the word ‘Group’ when they mean set……. Sets are also a great way of teaching multiplication & division at Primary level without having to introduce a whole lot of ‘buts’ and ‘ifs’ (Yeah – I know the standard definition of multiplication is ‘scaling up one number/quantity by another’, but that makes me want to put my head in a bucket of water, let alone an under-17)

Oh Stevie – if you ever read this – now I’m retired, I work part-time as an exam invigilator. Conversation I’ve had more than once ‘Miss, my calculator isn’t working’ Holds up calculator showing ‘Error’ message. Me – keeping VERY straight face ‘I’ll get you a spare’ Move to other side of hall so as not to see student’s face when same thing happens cos calculator will not do sqrt of negative number.

Hard to disagree with any of that! Some basic set theory has come onto GCSE but only really basic, I would like more. I’d also like to see Groups make it onto the A level syllabus.

per se – sorry!!!

My younger (8 & 10yrs younger) sibs had the privilege of doing the ‘New Maths’ GCSE – remember that. Oh – it soon disappeared – heaven forfend we should teach future zero-hours-contract workers to think and reason.

Groups at A level – absolutely – if you’ve done maths A, you should at least know what the damn things are.

PS I’ve also taught basic matrices to 10/11yo’s – it’s how you tell ’em. Isn’t CPA on the ‘decision making’ A level module (and yes – it could easily be on GCSE – but from my experience d m – which has more ‘practical application in the real world’ than most is interpreted by Sec Schools as ‘A level maths for dummies’

I teach Btec Engineering and it takes so long just to introduce the concept of calculus, let alone the application. My student are post 16 mainly.

We are forever using rates of change within mechanics and because of this in their first year of tech, they have three units which require calculus, so that’s good.

Now, if these students had been introduced to differentiation in year 10/11, well that would speed things up…so to speak (no pun intended).

Even integration, in simple form would help them so much, they do try to graphically work areas out by parts but that’s fine at first.

So bring on the GCSE calculus unit!!