We’re going to need a bigger boat

Here are my initial reactions to the new GCSE subject content document published on 1 Nov 2013 by the DfE.

So far this document is all we have.  We don’t know what exams will look like, how the tiers will work, and how each statement will be interpreted (for example a “simple proof” will mean different things to different people), so much of what follows is necessarily speculative. 

At the moment it seems sensible to be aware of some of the implications (like the need to talk to SMT about an increase in mathematics teaching time) but not to do any detailed planning (like rewriting schemes of work) until we have more information.

New material

First of all, there are new topics.  Some of these are clear and obvious (such as finding the equation of a particular tangent to a circle centred on the origin), while others can be interpreted in a number of ways.  

An example of this is the way gradients of lines will be used.  The word “calculus” is not mentioned, but at the higher tier pupils are expected to “interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents)”. 

I hope this means that essentially we will need to teach the concepts behind calculus (that the gradient of the chord is an approximation to the gradient of the tangent and that as the length of the chord shrinks the approximation is likely to improve) rather than telling pupils the way to differentiate polynomials.  A potential exam question for this could give an unnamed graph and ask pupils to draw a tangent at a particular point and then find the gradient of this, with an appropriate level of variation of the position of the tangent line being permitted.

Other queries include whether function notation will be needed, whether estimating a square root is as straightforward as noting that sqrt(20) is between 4 and 5 (or whether an algorithm is required) and exactly what is meant by “find approximate solutions to equations numerically using iteration”.

So far this is the DfE document.  The next stage is for the awarding bodies to create specifications and these will presumably include guidance that exemplifies topics like these one.

Beyond the headline new topics we will also need to ensure pupils are happy with using the kinematics formulae (which will be provided in exams) and that they have learned the other formulae they will need (because these, including the quadratic formula and the non-right-angled trig formulae, will no longer be provided).

Foundation will include harder topics

Aside from this there are a number of topics that will now fall into the Foundation tier.  This includes solving quadratic equations algebraically by factorising, and using the trigonometrical ratios.

Assessment Objectives

Assessment Objective 1 (AO1) is headed “use and apply standard techniques” and will be weighted at 40% of the higher and 50% of the foundation papers.  Much of the content in the document will fall under this heading.

The remaining marks on the papers will be equally split between AO2 (“reason, interpret and communicate mathematically”) and AO3 (“solve problems within mathematics and in other contexts”).  In the notes about AO2 and about AO3 there is the statement: “Where problems require candidates to ‘use and apply standard techniques’ … a proportion of those marks should be attributed to the corresponding Assessment Objective.”

I assume this means that a problem that involves reasoning within the context of Pythagoras’ Theorem will count as part of the AO1 marks (for Pythag) as well as part of the AO2 marks (for reasoning). 

This means that pupils will need to be comfortable with the problem solving and reasoning sections of the specification to be able to get all of the content marks.

What will the exams look like?

One of the key differences may well turn out to be in the structure of the exams.

A press release from Ofqual said this: “Exams only in the summer, apart from English language and maths, where there will also be exams in November for students who were at least 16 on the preceding 31st August.”  Finally, this gets rid of early entry completely.  

It appears that the highest attaining pupils will be stretched within the GCSE rather than needing to consider an extension course.  We don’t yet know what each grade will look like, so it is not clear whether the old G grade will map onto the new grade 1, up to grade 8 being the old A*.  I think this is unlikely because here is the only opportunity to recalibrate the system, so there will not be a direct link between old grades and new ones.

“Maths: Tiered with an improved overlapping tiers model. A foundation tier will cover grades 1-5 and the higher tier will cover grades 4-9. Assessed by external exam only, as now.”

If you just look at the tiers there doesn’t seem to be anything different about it.  Currently we have grades C-G (the lowest five grades) on the Foundation tier and the new version will cover the lowest five numerical grades.  There is currently an official overlap of two grades with the higher tier (with a compensatory E grade available for those who just miss a D) and the new system will have a two grade overlap. 

To describe the new tiering system as an “improved overlapping tiers model” must mean that model is improved.  Does this mean that the exam papers will be structured differently?  The document has some items in ‘standard’ type (for all pupils to know fully), some underlined (for all pupils to use and higher pupils to be more confident with) and some in bold (for higher pupils).  An early version of GCSE mathematics (when I took it in the first year it was available) had a common paper for all pupils and a lower paper and a higher paper (were they called Foundation and Higher back then?).  It will be interesting to see whether we return to a model like this.

If this is the case then we will need to have an increased focus on the higher tier topics.  At present the probability that any given different topic will crop up in a GCSE paper is incredibly small.  Half of the higher tier papers currently have to be questions at grades C and D, so A* questions topics can only be tested on about a sixth of the paper.  The new system might well give us more A and A* questions (sorry, grade 7/8/9 questions) so the pupils will need to study this higher content more thoroughly.  (As an aside, at the moment some students start AS-level mathematics not having previously been taught ‘completing the square’, or ‘rationalising denominators’ because they are unlikely to crop up in a GCSE exam.  An increased focus on the more difficult material in exams would make it less likely that this would happen.)

Teacher support

This will be important on a number of levels.  For those fully conversant with the mathematical topics involved there will be the need to flesh out exactly what is meant by each statement (such as the gradients referred to earlier).  For those who have taught mathematics in school up to GCSE level there may need to be some refreshing of subject knowledge for the new topics.  Teachers who are not maths specialists but who have taught GCSE (either at both tiers or only at foundation tier) may need to learn some new material. 

All mathematics teachers are likely to need to consider some new pedagogical ideas, because even the topics that currently appear at A-level will have a different emphasis in the new GCSE specifications.  (For example, we do not currently have long at A-level to delve into the meaning of the gradient and the new GCSE topic will afford us this opportunity.)

Teaching time for mathematics

In a written answer to the House of Commons, Michael Gove said:

“The new mathematics GCSE will be more demanding and we anticipate that schools will want to increase the time spent teaching mathematics. On average secondary schools in England spend only 116 hours per year teaching mathematics, which international studies show is far less time than that spent on this vital subject by our competitors. Just one extra lesson each week would put England closer to countries like Australia or Singapore who teach 143 and 138 hours a year of mathematics respectively. We announced on 14 October that mathematics, alongside English, will be double weighted in secondary school performance measures from 2016. This will also provide a strong incentive for schools to ensure that they are strengthening their mathematics provision.”

It will be important for mathematics departments to have the teaching time they need.  This will be important for the teachers, so they have sufficient time to cover the material (including AO2 and AO3), important for the pupils so they are able to achieve well on this new, more difficult course (if they don’t achieve a good enough grade the pupils will need to continue to study mathematics after Year 11), and important for schools because the maths grade will be double-weighted in performance tables.  (Incidentally, this double-weighting will start in 2016, before the new exams come into play.)

Where will this lesson that Mr Gove suggests we need come from?  Which subjects will lose out?  I am afraid I don’t know, but what I do know is that it will be important for the mathematics teaching to be done well and the extra time will be necessary for this.

It is also worth bearing in mind that it appears this extra period per week should happen in every year group at secondary school and not just at KS4.

Implications

These changes are for first teaching Sept 2015, so the current Year 8 will be the first year group to take the new exams.  Presumably it would be ideal for next year’s Yr 7, 8 and 9 to have an extra lesson per week. 

Are there enough mathematics teachers for the country to be able to staff this?  Probably not at the moment, so schools will want to appoint new teachers fairly early next term.

Scheme of Work

Much of the language in the KS4 document echoes language in the new KS3 national curriculum.  We will want and need to update our scheme of work to ensure it will still be appropriate not only for Sept 2015, but also to get the current Yr 7 and 8 pupils to 2015 with the requisite background knowledge and skills. 

If feels sensible to have an updated SoW for KS3 in place for Sept 2014, but then to wait until the final GCSE specifications and associated document (exemplification, sample papers, etc) are published and approved by Ofqual before making major changes to the KS4 SoW. 

Summary

So, we are going to need to start thinking about some of the implications (such as timetabling for next year), and will need to consider our schemes of work, etc.  We know we will need a bigger mathematical boat, but will also need to be patient and wait for further information before we can do all of our planning for the new GCSE.

Finally, I haven't yet mentioned that I like many of the new features in the document.  For example, being able to explore the background to calculus will be important and interesting.  It is good that AO2 and AO3 will maintain the focus on reasoning and problem solving.  Having more time to teach mathematics well will be welcome, as will be the increased importance of the subject.