Has the Higher level Mathematics examination become easier?

It is difficult to examine carefully the GCSE Mathematics examination without appearing to sound critical of the pupils studying for their GCSE, or sounding like you are questioning their outcomes as being possibly unequal to those from past years. Saying that, there is much debate regarding whether the GCSE Mathematics examination is easier than the 1980s ‘O’level or even whether it is now easier than the GCSE from the late 1990s.

Here, I intend to cast a critical eye over the similarities, differences and changes that the qualification has undergone. Then I’ll examine a few questions from each era and leave the final decision as to whether GCSE Mathematics is now easier up to you.

Firstly, we must address the point that ‘O’levels were not intended for everyone. Many pupils took the CSE examination, which was intended to be easier and thus more accessible. By the mid-1980s, around 20% of the school population were taking ‘O’level Mathematics with the other 80% either taking a CSE or leaving without a mathematics qualification at all. We shouldn’t forget that a grade 1 at CSE was regarded as equivalent to an ‘O’level grade C, though many people still viewed it as an inferior qualification.

After Conservative Education Secretary Sir Keith Joseph decided to proceed with a merger of ‘O’levels and CSEs, the first GCSE courses began in 1986, and the first examinations were taken in 1988, though we should not forget that there had been trials previous to this full implementation. The central idea was that most subjects should be examined through tiered papers focusing on different parts of the grade scale, ensuring that each grade reflected “positive achievement” on appropriate tasks, rather than degrees of failure. As is currently the case, a grade of G is a pass, but that was quickly superseded by the idea that a grade C was a pass and anything lower should be considered a fail. GCSEs had the best intentions, allowing all pupils to learn the same syllabus and attain a grade C or better. Unlike a grade 1 at CSE, a grade C on a Foundation paper was indistinguishable from a grade C on a Higher paper. This though, was not the case with GCSE Mathematics.

For a long time (1988 to 2008) Mathematics was a three-tiered examination: Foundation, Intermediate and Higher, with respective grades G to D, E to B and C to A (later A* when it was introduced in 1994), though we should not forget that grade E at Intermediate and grade C at Higher were both meant to be “compensatory”. Now there is no doubt that under that system pupils taking Foundation were disadvantaged, as they were unable to attain a grade C, but it had its advantages for those on the Intermediate level, as they were able to attain a grade B without needing to study the most challenging parts of the curriculum.

Under the current two-tiered system: Foundation and Higher, the grades are G to C and D to A* respectively, with the Higher examination grade C now being possible to attain with around 30 to 35% of the total possible marks, leading to inevitable questions regarding the dumbing down of the GCSE. Unlike with the three-tiered system, a Higher paper now has approximately 50% of marks available for questions that are roughly equivalent to grades C and D, with a little overlap into grade B; though care must be taken when attributing grades to questions as many have such exercises are based around previous examination structures which have been superseded. So, given that grade boundary for a B hovers at around (or just under) 50%, we again see the ammunition those who claim that the GCSE has been dumbed down. In effect, a pupil doesn’t need to know any “true” Higher level mathematics in order to attain a grade B! Worse still, they can start an Advanced level course without having the proper GCSE foundations of study required for proper mathematical advancement. When a three-tiered system was in operation, only a tiny minority of Intermediate level students started on an Advanced course. We must therefore question whether the switch has hindered mathematical progression by setting unrealistic targets for students regarding their ability to study advanced materials.

What remains clear though, is that throughout the life of the GCSE, the syllabus has been mostly constant. I would estimate that some 99% of the syllabus from 1986 can still be found on the syllabus from 2010. Which then raises the inevitable question as to how examination questions have altered?

Most newspaper articles on this topic are disingenuous, using either questions from different tiers, or questions from different parts of the paper. What is more useful, it to look for similar types of questions and ask whether they are of similar difficulty.

Take this question from 1984’s ‘O’level paper:

Solve the simultaneous equations    3x+2y=-4   and  x-3y=17

And compare it with that from 1999:

Solve the following simultaneous equations  8x+3y=35      and   2x-5y=3

Finally, with that from 2011:

Solve algebraically these simultaneous equations  3x-y=1   and  5x+3y=4

All, I think you’ll agree are quite similar. Yes, that from 1999 requires a multiplication of both equations before elimination, but the other two have answers where one unknown (or both) is either a negative or a fraction. In many ways, pupils would find the question from 1999 the easiest, except that the numbers involved become quite large.

What is apparent though, is the differing levels of algebra. For instance in the 1984 paper we see:

Factorise  1-p-12p^2   and    Solve   2x+3=4(x+1)  amongst others.

At (around) the same question number in 1999, we find:

Find the solution for the following equation 5(a-3)=3a-19  and rearrange to make y  the subject of x=120y+136.

And in 2011 we find:

Solve 3(2×-5)=30  and   Write as a single power of  p     p^2  x p^6      and   (p^2)^6

Now, as any mathematics teacher will tell you, having a proper foundation in algebra is a pre-requisite for studying at Advanced level, we can see that these three are not similar in difficulty. Now, that is not to say that later in the paper we didn’t see more difficult algebra in 2011, e.g.

Factorise  10x^2+5xy    and Factorise and solve  2x^2+7×-15=0

But the question remains as to whether a pupil gaining a grade B will need to have answered such questions correctly to attain that grade, especially as a grade B is seen as a gateway grade for study at Advanced level.

Overall, whilst the three papers are quite different in style, many questions are similar in the topics they are testing. It is only the number at the “higher grades” that is inconsistent, with many more in the ‘O’level paper and 1999 paper than in the 2011 paper.

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Has Michael Gove done anything worth praising?

I take this serious idea from a blog on the Daily Telegraph site. Has he done anything really worth praising? Are his ideas sound and worthy of praise? Or are they idelogically flawed? Lets take the changes made by or soon to be made by the Education Secretary one by one.

Free schools.

Introduced with the idea of creating parental choice, we must first ask ourselves what is the level of cost associated with this idea? Toby Young the advocate of free schools wrote that the DfE was going to spend another £600m of free schools. At the same time, the NUT is finding that free schools are not serving everyone, only those parents that we can easily classify as “middle-class”.

More worrisome is that free schools are free from the National Curriculum, which the Education Secretary is determined to remove in Secondaries anyway (more on this later), meaning that Religious ideology can be the mainstay of a free school education.

Academies.

Introduced under Labour, these have been the biggest reason for the issues that are now spiralling out of control in education. Taken out of LEA control and influence, many academies removed pupils from academic courses purely to boost examination results. Pupils were placed on courses that gained them the “equivalent” to up to 4 GCSEs, yet were seen as having little or no value by employers.

When league tables were amended to state that pupils having 5 GCSEs grade A* to C including English and Maths, these schools were initially found wanting, then amended their curriculums to increase the time spent on teaching these core subjects to the detriment of a more rounded curriculum.

Have academies increased outcomes beyond those schools that were still under LEA control? Not as far as the statistics would suggest. In fact, when the EBacc reporting was added to league tables, some academies were the lowest performers.

It is easy to argue that academies have decreased choice for pupils, not improved it. Increasing the numbers of academies only further decreases choice.

The Reformation of GCSEs.

The initial proposals for reforming GCSEs looked ambitious and purposeful. There was the thorny issue of a two-tier examination system to be resolved, but the amendments to those intial plans seeked ways to overcome this. They argued that the present GCSE was already a two-tier system, having Foundation and Higher levels for most subjects; which is quite right, they do.

What is not so apparent is how these new qualifications are going to be an examination to stretch the brightest, yet be accessible for all. Will certain subjects that are most easily determined by types of learning outcomes, such as Maths and Science, where they are best suited to examining skills use to answer problems or explain systems. It is far harder to examine and grade these subjects through developed skills in answering the same problem, than it is for Geography or History. So, do we create a totally diferent style of examination for these subjects, or have swathes of questions that don’t test the most able and are only there to fulfil quotas?

As the Education Secretary appears to see these examinations as a return to O level style exams, we could expect the latter, rather than a new style of examinations. The only alternative would be that large numbers of pupils never attempt or expect to pass theses exams and that is a far worse position than we currently find ourselves in.

The New National Curriculum.

Again, this was an ambitious endeavour that looked promising in its scope. At a time when SATs results had plateaued, it seemed a far idea to find a new stimulus for improvement, but then this year’s results arrived and there has been considerable improvement. Is a new NC required at all? Or doesn’t Michael Gove and his colleagues trust the KS2 SATs results and eyye them with mistrust in much the same way that secondary teachers do?

Then there is the new phonics test. I think I’d have failed, unless primed to not question non-words. I was a good reader and like anecdotal evidence suggests, would have questioned certain words as either spelt wrongly or possibly I’d have asked what they mean. Such testing suggests that educationalists advising Mr.Gove don’t trust teachers to teach reading skills, even though test results at KS2 would suggest the opposite! Yes, it is a scandal that 15% of pupils don’t reach the expected level, but this would be improved by careful intervention methods and indentification of causal reasons, rather than a blunt instrument like phonics testing.

By all means strengthen the NC for primary schools, but is a complete overhaul required?

In Summary.

To me, these are the four main areas that Michael Gove has endeavoured to change. Many have had the right intentions, but appear rushed or just plain pandering to the “chattering middle-classes”, i.e. Free Schools. Grade inflation at GCSE has been excepted by politicans of all parties, so here was a real opportunity for creating a cross-party system of new examinations.

At the moment, his policies appear too rushed and poorly conceived. His denigration of teachers and appointment of Michael Wilshaw at Ofsted appear as nothing more than an attack on those than are most required to implement these changes.

It is impossible not to find that Michael Gove is failing in his duties towards our children.

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Should teachers be granted anonymity against claims of crimes against pupils?

http://www.guardian.co.uk/education/2012/sep/28/teachers-accused-crime-pupils-anonymity

This debate has become more important to many, given the recent media attention on Megan Stammers and her Maths teacher, Jeremy Forrest.

Teachers accused of committing criminal offences involving children at their school will be granted anonymity under regulations which come into force next week.

Whilst many accusations are found to be malicious, there are a number that are not. The difficulty comes because parents place a great deal of faith in teachers keeping their children safe. Once that faith is broken, the teacher is no longer capable of teaching in the same way towards those children.

There have been a lot of comments online regarding whether Forrest should be treated differently from “the normal man on the street” with regard to having a relationship with a girl of 15. Now, before I address the morality of a man of 30 having a relationship with a girl of 15, we should remember that parents place a lot of faith in teachers as stated above. Forrest was trusted by Megan Stammers’ parents and he has broken this trust. What lasting damage it will have done to the girl we’ll probably never find out. Simply put, Forrest was not in the same position as “the normal man on the street”, he was in a position of authority over Megan. Anyone who has taught has experience of the near awe that some teachers are held in, with regard to pupil’s opinions. Forrest was a more learned person, with a “cool” hobby; music. To a girl of that age he would quite likely seem sophisticated and powerful. Many women mention having had crushes on teachers whilst at school. This is normal and a natural part of the maturation process. Teachers should never have a relationship with a pupil, as the current education act requires them to avoid.

Morally, many are like myself and find it quite disturbing when we hear about a girl of 15 being in a relationship with a man twice her age. There is a level of near abuse in such a relationship due to different life experiences, as well as emotional maturity. Quite simply, such a relationship is morally wrong in any environment.

This all makes anonymity in regards to allegations a difficult decision. Even a false allegation of improper behaviour towards a teacher can wreck a career. At the same time, the protection of children should be upper-most, surely. So, where should the line be drawn? If the legislation is to work, systems need to be in place for schools to act swiftly and the police must decide whether to charge equally swiftly.

My trouble with this legislation comes from the accusation that under this act,

The accused teacher could be disciplined, sacked and move on, but unless they were charged with a criminal offence, no one could name them and the new school might not be allowed to know there had been a problem,…

As the Crown Prosecution Service might not have enough evidence to charge, allied with often poor motivation by the police and social services towards seeing under-age girls in a sexual relationship with an adult as being a victim, means that that if Satchwell is correct, a loop-hole exists that could mean a teacher who has been in a sexual relationship with a pupil could just move schools. Then possibly re-offend.

Most legislation is far from perfect, but you have to ask whether this one was rushed without proper indentification of the possible issues and their solutions.

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Do rigorous examinations have to be memory tests?

I must admit this is the biggest misrepresentation in the examinations debate. As a Mathematics teacher, it would appear that often a Maths exam turns into a memory test, and this is the argument used most often by those who are condemning rt.Hon Michael Gove’s plans.

Now, there is no doubt that any Maths exam will have a large element of testing the memory. There are many different methods used in calculating the answers to percentage questions. Yes, each can be derived from the basic method,  but the pupils need an understanding of how percentages are calculated in order to derive the methods. The examinee needs to carefully read the question and then determine the course of action required to calculate the answer.

This topic is not the only one where a well written examination question can test whatever is intended. Some people think algebra is best tested by simply creating questions that test the recsall and use of recalled methods. That is not all though. It is entirely possible to create questions where the pupil must use algebraic techniques to solve problems. In fact, it is quite possible to create a very rigorous examination that checks understanding as well as memory (recall).

To that extent, it depends upon the marking criteria as to whether an examination ends up a memory test or an examination of a person’s ability to use information and reflect upon it. Take History for instance; why is it difficult to have a pupil read and interpret an essay or part of a longer structured research article. It is how the answer is marked that makes the examination rigorous.

So, as I’ve said, the whole debate surrounding rigorous examinations being memory tests is nothing but an insulting misrepresentation.

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Education is in need of direction.

As the English examination situation rumbles onwards, with the Welsh Education Minister attempting to deflect the poor standard of results in Wales by demanding the regrading of pupils, it becomes ever more apparent that education has lost it’s purpose.

As the Rt.Hon Michael Gove’s plans for a new examination are leaked to the Daily Mail and then quickly picked up by news services, we await to see the details but must already wonder about their direction when he appears to suggest that “top grades” will be rationed. This looks as if some form of “norm-referenced” grading will become effective.

Now, I understand that many people support this system but it is distinctly unfair. Under a norm-referenced grading system, widespread improvements to education standards will not be rewarded. Outcomes would improve, but their rewards would be withheld.

What is required are rigorous standards for examinations, carefully marked to the highest standards with criterion-based outcomes. It is entirely possible to create examinations that are not identifiably “expected” in the way that many Humanities examinations appear to have acquired a system of “rolling topics”…e.g. last year was X so this year will be Y.

But, at the basis of any educational change should be a clear intention and that appears to be missing.

Is the Rt.hon Michael Gove’s intention to create a series of examinations that test learning with the purpose of dividing pupils into those capable of higher education and those that or not? Or is the intention to create a skilled workforce for employers? Is it possible to do both under a single “umbrella” examination?

When one reads what skills employers demand (when they agree or refrain from their normally schizophrenic stance) a general certificate of schooling would suffice, rather than school-leavers having a range of examination passes. Generally, they demand a reasonable standard of literacy and numeracy along with “softer” skills such as team-work, adaptability, listening and creativity. None of latter can be examined by a system as forwarded by the Education Minister.

Does a system so clearly based around his early criteria for league table success, the English Baccalaureate, actually create a workforce skilled in the ways desired? Does the system appreciate a need for vocational qualifications and give them a form of acceptance? Does it want vocational qualifications to have any acknowledgement of viability even?

So, I ask in what direction is our education system aiming for?

Until it knows we’ll forever be looking for the next fix to a problem that can change almost daily.

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