Irrational numbers

In this case, I am not going to talk about real numbers which cannot be represented by an integer fraction or a recurring decimal.  Instead I want to talk about normalized percentages.  “What makes these irrational?”, I hear you say.  Their application to examinations and assessments is what.  During the past week or so, two sets of examination results have been released in the UK – the A-level (two or three of which are usually required for entrance to university) and the GCSE (usually seen as a ‘basic’ qualification in a subject if grade C or above is obtained).  These results largely affect England, but many schools in Northern Ireland and Wales, and a few in Scotland, also take them.  For my non-UK readers, the four countries making up the UK each have their own education systems – you can find an overview in the #mathchat Wiki!

The A*

The big change for students completing A-levels this year was the introduction of the A* – an attempt to increase the value of the ‘gold standard’, as I see it.  At the time of its introduction, there were some who saw it as being pandering to the elite, however.  The history behind the A-level and GCSE is well summarized in an article from  Incidentally, I find this is a good site for information/news related to politics in the UK, since it’s articles are written to provide useful references as well as breaking news, and tend to be more balanced in their points of view.

To get an A*, you must meet the following two requirements:

  • 80% UMS overall (the current requirements for an A),
  • 90% UMS overall in the A2 units.

UMS is the uniform mark scheme/scale (uh-oh, here be dragons!).  You don’t get an A* for 90% overall. Only the A2 units (i.e. the units normally taken in Year 13, the final year) count towards the award of an A*. Even if you get 100% in your AS units, 89% in your A2 ones isn’t enough for an A*. Similarly, you don’t need to get an A at AS to get an A* at A2 if you get 90% in the A2 units, and assuming your A2 marks are high enough to pull your AS marks to an A grade overall.  Unless of course you are taking Maths, Further Maths, Further Maths (Additional), Statistics, and Use of Maths AS. No AS will have the A* grade available.  Although some of the individual units in the Edexcel A-levels have been modified; see here [The Student Room, an excellent online student resource in the UK] .  You can get some idea of the complexity of what is involved by looking at the course structure for Mathematics, for example.

Ho hum….

The A* itself is being awarded to help universities (and others) distinguish better between candidates who had three As.  Some universities, for example  Cambridge, announced what their standard offers were going to be: in some cases for most subject entries, in other cases only for specific degree subjects.  Most of the subject entries, at a quick glance, seem to require mathematics, and some universities specify that the A* has to be in A-level Mathematics (naturally enough if you have a reputation for science education such as Imperial College).  Imperial College, however, has also stated that it will be introducing a university-wide entrance exam for all applying to study there from 2010, because “We can’t rely on A-levels any more.” [Sir Richard Sykes, Rector of Imperial College quoted in The Sunday Times].

“We are going to have entrance exams that will test ability. We are looking for students who really will benefit from an [Imperial College] education. The examination will look for IQ, intelligence, creativity and innovation and will not be too dependent on rote learning.” [Sir Richard Sykes, ibid]

As far as the A* itself is concerned, there is a big “but”.  The grade is only being awarded to people sitting their final A2 exams from September 2009 onwards (the grade first being available in the summer 2010 certification just past). This means that people who took their AS-levels in summer 2009 or before will not be able to get A*s.   Except, of course, that if one of your A-level modules was Use of Maths, which is an AS and not eligible… dum-di-dum.

I hope you’re keeping up, you’ll be tested later!

I have one word which I feel would address most, if not all, of these issues: portfolios.

GCSEs and the bell-curve

First of all, it would be unfair to proceed without providing a link to the official summaries about the GCSE (sanitized for you protection, of course).  There is, of course, a UMS for GSCEs as well.  This is the UMS for GCSEs from AQA, which is the largest of England’s three examination boards.  OK, first question: in a 30-page book explaining uniform marking (AQA‘s), would you be a little concerned if 19 of those pages were tables of boundaries for various different grades and subjects?  As a statistician concerned with validity of data collection and comparison, no.  As a teacher, particularly of Mathematics!, yes.  Second question, do these tables as they are give a clear picture of where the grade boundaries lie?  Maybe not… let’s try a chart using Table 6 from Appendix A of AQA’s leaflet which shows GCSE Modular Mathematics (Specification B) (4307) two-tier without coursework.  You can make your own chart if you want something else!

Minimum mark required to get a particular grade in GCSE mathematics.

Basically, grade C is considered as a pass.  This is a two-tier GCSE and a pass level is available in both tiers.  In this case there is no coursework element.  Coursework is usually marked by teachers in the school using a preset scheme and then verified by external examiners. The N grade is only given in the higher tier to those who fail to achieve the minimum required for a grade D.  No account has been made here of the weight carried by each of the three modules and their contribution to the UMS score.

A couple of things stand out to me now, with the visual presentation of the data:

  1. The UMS is clearly aligned to the marks in the higher tier modules, where all grades except for D are considered as a pass.
  2. Lower tier students (oops, I mean foundation…) are expected to work proportionately harder to achieve the higher grades than those doing the higher tier.

In respect of 1, this is hardly surprising since the idea behind the grades and uniform marking schemes seems to be that a particular chunk of marks of a standard size represents an increase in grade or level.  The word grade, after all, derives from the Latin gradus meaning a step or degree.  We don’t build staircases with uneven step heights, unless it’s absolutely unavoidable, so we don’t create grading systems with uneven steps either, do we?

In respect of 2, the bar heights for a grade C in the Foundation tier lie between those for A and A* in the Higher Tier, similarly grade D bars in the Foundation tier are slightly above those for a grade B in the Higher.  Now this seems inherently unfair, to me: to be asking lower-ability students to work harder than their higher-ability counterparts, who should be capable of working at higher levels and, perhaps, are not being sufficiently challenged.  The following chart shows the contribution of the modules by weight to the grades for both the higher and foundation tiers, which shows the disparity more clearly, I feel:


I suspect this disparity comes from ‘normalizing’ the Foundation level grades by using bell curves.  I could go into more detail, but Professor Miller does a good job of explaining this concept for one of her biology courses, so I am not going to reinvent the wheel.  The student grades are then probably normalized each year to make sure the appropriate number of students achieve passes at that grade, so making it more difficult to make comparisons from year to year… or am I pushing the boundaries beyond belief here?

Now, if I am trying to decide whether or not that person in front of me, with a grade C in Mathematics, is someone I think will be able to work for me or progress to the next stage of their education, I have a problem.  I don’t know whether or not they don’t know 40% of the curriculum or just over 30% of the curriculum. Or, did they do enough to get 60% of each question correct but not reach the final answer of any of the problems?  That’s a silly example, or is it?

The point is, a grade may tell me something about a student’s knowledge of a subject, but it doesn’t tell me what they can or, more importantly, cannot do!

I have one word which I feel would address most, if not all, of these issues: portfolios.

Anyone spotted a theme here… 😉

2 responses to “Irrational numbers

  1. For the sake of balance, or cause of further upset, I should also mention Edexcel’s UMS for the same GCSE qualification in Mathematics. If you do the calculation for the Foundation tier grade C, you’ll find it is 75%…

    Sometimes, I wish I wasn’t a mathematician who could calculate these things… *sigh*

  2. This post is one of many featured in Standards-Based Grading Gala #2. You should visit that if you are interested in formative assessment, and the US approaches towards and thoughts about standards-based assessment, which is new for many.

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