Thinking Aloud: Are There ‘Two’ Mathematics?

“They laugh at me because I’m different; I laugh at them because they’re all the same.”

Kurt Cobain

Imagine you decide to learn a new language. As you’re a bit keraaazy you decide you’ll pick something totally left field. A language that relies on little of that which is native to you; symbolically extremely varied; can be communicated (and this is a key point) in a variety of different notations, orthographies or diagrams and whilst is used commonly across all nations is misunderstood by the majority of the population.

You know, of course, that I am talking about Mathematics. Mathematics is a study, of number, geometry, algebra, statistics, etc.

Every single day – EVERY SINGLE DAY – we come across a mathematical construct of some sort: a 20% off sale; the gradient sign on a hill that we’re driving up; the reading on our weighing scales; how long the Sunday dinner will take to cook; these are all situations where we’re using Mathematics. We are constantly making sense of the world around us through the means of Mathematics.

Mathematics, therefore, is a language in its own right, a means of communication as well as a system of study. It is the language through which we make sense of the world around us at a technical level. We describe using words but those words are in themselves descriptors of the Mathematics we use.

Now for me, at a school level, are we teaching Mathematics, or how to communicate mathematically? Presently, for me, we are trying to teach both. Let’s use an analogy. English at Secondary level is taught as a language (the syntax, grammar, spelling etc.) and as a study of literature (prose, poetry, etc). The language part focuses on the rules and systems. The literature focuses on the application and interpretation of the use of the language. In the present Ks1-KS4 curricula for Mathematics, we teach everything together, both the rules, the application and interpretation, right the way through.

Now, I have my concerns about this. To carry on that analogy with English, most schools these days make sure all students study at least English Language GCSE, in the intention that they can at least communicate through speech and written form (well, in the main). Yet for some reason – of which I don’t know why, Mathematics students are not just expected to communicate in the language of the subject but apply and problem solve in it, all in one complete package.

Worse than this – and this really makes my blood boil – the content of the Mathematics GCSE is so wide-ranging, and examined so broadly yet only is worth one GCSE to the student. Now, there have been efforts to rectify this. The Linked Pair Pilot for example, was an attempt to acknowledge the subtle difference between mathematical communication and application and create a two-GCSE examination. Yet it’s being shelved.

I recall my PGCE tutor, a very philosophical man, once exclaim to us that Mathematics should be a GCSE option (!). I for one do not agree, but I do think that a huge chunk of the GCSE should be optional. There should be a compulsory Core Mathematics (Communication?) GCSE, and an Applications of Mathematics GCSE as an option for those students who are suited to going into technical fields.

“WOAH!”, I hear you cry, “ARE WE NOT GOING BACK TO THE DAYS OF GRAMMAR SCHOOLS AND TECHNICAL COLLEGES???”. Well, no. This is not some attempt to seperate the haves and have-nots at all. 

Mathematical communication can be very difficult at Secondary level, so there would be challenge for all taking such a qualification. Take Histograms, Moving Averages, Compound Interest and Depreciation, all can be quite tricky for students and would be part of a Core GCSE because they are used in communicating mathematical situations.

The other stuff – I’m talking Vectors, Circle Theorems, most of Algebra (but not graphs) should be placed in the Applications GCSE. Why? Because the Core GCSE should be about what people are going to come across in their everyday lives. There’d be a place for Functional Mathematics as well, not something tacked on as a token gesture but properly built-in.

“WOAH”, I hear you cry, “BUT ALGEBRA CAN BE USED FOR…” – yes, yes, I know. But there need to be a distinction between helping somebody to be mathematically competent enough not to be ripped off by and helping them get on to a Physics course at college. Plus what’s to say that the Applications GCSE can’t be taken at a later date? Would we be that cruel to say “unlucky pal, you had your chance at school” (shockingly enough that’s how a lot of people feel as adults – it’s never too late to change).

This approach, therefore, would allow more time to be dedicated to ensuring students are fluent in the language of Mathematics – the rules, the methods, the core business that ties everything together (fractions, decimals, percentages, ratio, proportion, rates of change, etc). We’d have students at Grade C (or 4, or 5, or whatever the government of the future decides next time) who could actually work out how much to put away each month to pay their bills. They’d know if that mortgage deal HSBC are offering is better than the one Santander are.

Meanwhile those skilled in Mathematics could go on and do a ‘proper’ Mathematics course not encumbered with the “Johnny sells teddy bears, if he sells 30% for £5, half for £2 and the rest at £1, how much money does he make” questions but with the good stuff – the stuff they’d see in the Level 2 Further Maths Certificate, a bridge to A-Level Maths and Physics instead of a distant third-removed cousin. There are challenges to this; indeed actually getting students to take Applied Mathematics would be difficult enough (it’s tough enough trying to get students on board for a compulsory Mathematics GCSE!), it’d need real backing from colleges, universities and the government, plus yet more teachers to fill the gaping void that is the class of ‘quality teachers of Mathematics’. But who cares? We face those challenges now anyway, don’t we?

The results of such a change would be plain to see. We’d have more time to concentrate on getting students to master arithmetic and the truly day-to-day applications of Mathematics. We wouldn’t have to rocket through the curriculum, trying to cover every topic in the hope that they’ll not be caught out in the exam (hands up if you rush through teaching vectors at the end). Those who show a flair for Mathematics would be able to actually take an additional qualification which would give a grounding in the concepts required for further and higher education.

What are the chances of this happening? Well, I think quite slim, if I’m honest. The government, Ofqual and the like had a golden opportunity to make this happen, and whilst the awarding criteria mean that GCSE Mathematics will soon count twice for school performance figures, in all reality it’s still only one GCSE. I think this is madness, considering the sheer volume of ‘stuff’ that students need to study in order to have any chnace of success.

There are (teacher led) moves to address this problem – mastery curricula, for one, where the concept of getting the ‘core’ right first before moving on to the ‘application’ is the central principle. I’ve put my hat in the mastery ring for this exact reason. There’s nothing more frustrating than trying to cover simultaneous equations when students don’t understand the basics of inverse operations, which is a result of a misunderstanding of the opposite nature of addition and subtraction (and for that matter, multiplication and division).

A final point. Imagine going to a different country – Iran, say – and not learning the language (Farsi, if you’re interested). You can’t expect the locals to speak to you in your language, so you’re stuck. How on earth can you expect to succeed? I’m not even saying reading a novel in the local language – just being able to talk to another person. In fact, imagine all of this, and then be expected to do a test on said language. It’s impossible, no?

That is the situation that many of our students (and adults) find themselves. We cannot allow this to continue. We must focus on the communicative, linguistic base of Mathematics and get students to understand that before we actually get them doing Mathematics. I feel that the current qualification structure does not allow this to happen, and as such, we are failing – yes, failing – a huge section of the population.


About workedgechaos

Teacher. Critic. Geek.
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3 Responses to Thinking Aloud: Are There ‘Two’ Mathematics?

  1. “But there need to be a distinction between helping somebody to be mathematically competent enough not to be ripped off by and helping them get on to a Physics course at college.”

    Well there is a distinction, but I often wonder if it’s one that’s about anything more than aspiration. Teachers wanting their own kids to learn maths well enough to study physics at college while saying others people’s kids just need enough to manage a payday loan. It’s easy to find problems with the way maths is done in England, because it isn’t really done very well, but so many “solutions” are not about doing it well and are just about not doing it, or not doing it with “kids like these”.. I find this to be the case particularly with algebra. Is it really that algebra is useless, or is that we teach too little of it for kids actually to be able to use it in real life?


  2. Pingback: Thinking Aloud: CPD, We Hardly Knew Ye | Teaching at the edge of chaos

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