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Fifteen thousand students were affected back in June by mistakes contained in a question on paper two in Higher Level Maths. With results due out tomorrow morning, and a report for Education Minister Ruairí Quinn nearing completion, teacher Eamonn Toland – managing director of TheMathsTutor.ie – reminds us what happened, and outlines how the State Examinations Commission can make sure it never happens again.
THE MAIN ISSUE was that the trigonometry question was set out with too many items of information, thus creating a scenario that was not mathematically possible. In effect, examiners attempted to describe a triangle using a set of measurements that can’t actually co-exist in a single triangle. This meant that the question was fundamentally flawed. Students answering the first question posed about this scenario could arrive at different answers using legitimate mathematical methods. Follow-on questions would also be impacted. (For a video overview of the problem, click here).
So, how does this affect students?
Well, the stronger students would have found it unusual to be provided with four measurements to describe a triangle. Of course a triangle has three angles and three side lengths, but usually you’d expect to be given two side lengths and an angle, or two angles and a side length, and that would be enough to figure out the other measurements. Being given four measurements should set off alarm bells, especially for a strong higher level student.
Because of this, students might have chosen to solve the first part of the question using very basic trigonometry, and have then decided to quickly check their answer by using the same method but going at it (literally) from another angle. If they did this, they would be very puzzled, because they would arrive at two different answers. They might waste time trying to understand what was going on, and it could throw them off their stride.
Furthermore, the answer to part (a) feeds into part (b), so this student would be feeling very uncomfortable with the dilemma as to which answer was correct.
Not all students would be affected. Some might not realise the implications of being given four measurements instead of three, and would proceed to find the answer to the first part, and not do any further checks. They could then proceed to use their answer in solving part (b).
Note 1: the Irish language version of the paper did not include one of the measurements, namely the 36 degrees angle at H. Because of this, their question 8 was legitimate. However, this ‘omission’ was noticed, and ‘corrected’ verbally by means of an announcement by examiners before the exam began. This meant that Irish language candidates then became embroiled in the same problematic scenario.
Note 2: There’s a second error in the question, in that an exterior angle is effectively described in the text description as being 124 degrees, but it is the interior angle which is labelled as such in the diagram. It is highly likely that most students would have gone with the diagrammatic description, but it’s still not acceptable to have this kind of error, especially as different students have different ways of processing information. This could have been another source of confusion.
The SEC could do one of several things:
There are problems with all of these approaches. Too ‘soft’ an approach is unfair to students who are genuinely strong on trigonometry, as it effectively levels everyone’s marks for this question. Too ‘hard’ an approach could penalise all students unfairly, including the very best students as mentioned in scenario (c).
None of these approaches can compensate the students who spent extra time on this question, or who lost momentum or confidence in the exam due to this question.
The SEC has stated that the official marking schemes will be available by the end of August, in time for students to view their scripts. However, they may release a statement before then, due to the notoriety of this particular incident.
Anecdotally, there has been a mixed reaction: Most students, parents and teachers have been rightfully upset that this question could be set with these significant errors, not to mention other errors seen in other exam papers, including maths, this year. A lot of students have said that they hadn’t realised that there was an error in this trigonometry question, and as such were unaffected on the day. However, they are still concerned about how the question will be marked.
Some other students experienced unease at the question having too much information, but in the exam situation, took it at face value and did not necessarily investigate too deeply. Others did the check and became confused when they got two different answers. This caused a delay and upset during their exam.
How can this be prevented from happening again?
Exams are drafted and set by a team, including at least one drafter and one setter, and possibly assistant setters. These are answerable to a chief examiner who has ultimate responsibility for setting the exam. The setting process includes proof-reading, review and revision in accordance with SEC procedures.
Human error is always possible in any endeavour. Although the exams are set by a team of subject matter experts, it’s important that the quality assurance process be well designed, and fully complied with.
It is difficult to understand how so many errors have slipped through this year, particularly in maths. To avoid this in future, there should be more clarity and accountability in the process.
A report is being prepared for the Minister regarding the mistakes made in setting exams this year. We hope that there will be no more such mistakes, particularly in maths, from 2014 onwards. Irish students must be able to trust that their exams are based on valid and fair questions ongoing.
Eamonn Toland is founder and Managing Director of TheMathsTutor.ie, and leads a team of mathematicians, teachers and technologists who have created Ireland’s leading e-learning system for Project Maths.
If the angle why (no-one copped it) is greater than the angle how (someone messed up in the first place), through how many degrees does a Department have to spin before the truth will out. Riddle me that. (360 marks)
This is just silly. Any problem with any paper will be solved by examiners at the marking conference. They are given the job of advantaging all students in whatever way they can.
Suggesting there is a profound distrust of the exam system is without foundation.
All articles like this do is wind students up on the most anxious night of their short lives
With all due respect Fintan, there is nothing silly about the litany of errors in exam papers this year, particularly in a number of maths papers. In particular, there is no excuse for having these 2 errors in the question which I discussed in this article.
The SEC will have been unable to resolve the problem at the marking conference in a way that is fair to all candidates – there are too many variables at play. They can make the best of a bad job, but it should never have happened, and they must ensure it never happens again.
To state that I am suggesting there is ‘profound distrust of the exam system’ is more than a bit exaggerated – I can’t see anything in the article to suggest that. But yes, confidence has been shaken, no one should attempt to excuse or play down what happened, and the SEC must shoulder the responsibility of rebuilding that confidence. I hope that all reasonable people would join in calling for assurances that procedures be improved, and that they be followed rigorously to avoid any repeat of this year’s level of errors.
Eamonn, it is silly to suggest that anything less than the best will be done for every student sitting a state exam. Have you marked papers? Five minutes in a marking conference would disabuse anyone of the myth that examiners are ‘out to get students’. There are thousands of variables in any exam system, mistakes are unfortunate but inevitable. They are always fixed.
The contention that confidence is shaken comes from the title of the article!
Most of all though, publishing/writing an article like this on the day before the results does students and their examiners a disservice.
Again, with respect Fintan, I am not suggesting that examiners who are correcting papers will do less than their best with what they have been provided.
Likewise, I am not suggesting that examiners are “out to get” students, so it is quite futile to imply that I think this, or that I am saying this. I can only respond to the content of the article that I wrote, not to any other ideas that are being projected on to it.
I do not wish to argue or quarrel with you, so if you don’t mind I will agree to disagree with you and leave it at that.
The issue (the problem, if you like) isn’t anything to do with examiners being out to “Get” students. It’s that a mistake like this on the exam paper complicates differentiating between the mathematical abilities of students. Sure, mistakes happen. The difficulty when mistakes happen is how significantly they cause an issue with illustrating that one student has a better command of the problem and the solution than another student has. A grade here, a grade there – for many students, that’s the difference between getting the college course they want and missing out.
Eamonn’s article makes a good point – it’s not as simple as distributing the marks around, ignoring the question, giving full marks to anyone who attempted it or giving full marks for the question to everyone. Each of those solutions causes a difficulty with one of the major reasons for doing the exam in the first place – ascertaining where on the ability scale each student is in the particular subject. It would be a disservice to the students to ignore the issue and refrain from printing the article. As for the examiners… well, I’d argue that it would be a disservice to them and to future examiners if the issue was ignored in the media. But in any case, they’re the ones providing the service rather than receiving it.
Looking at the results today proves my points above: students get what they deserve, a system with external anonymous grading is the only way to ensure our students achieve their expectations.
If there’s a guy in a rowboat going x miles…and the current is going, like, you know, some other miles, and how long does it take him to get to town? It’s like who cares? Who wants to go to town with a guy who drives a rowboat?! – Michelle (Of Romy and Michele’s High School Reunion Fame)
Anyone who got a sensible answer that did not break any trig rules should get 80% marks.
Any student that got a sensible answer that didn’t break any trig rules and spotted that the question is bogus should get 100% marks
Any student that spotted that the question is bogus should get 100% marks
This is based on the face that it would take more work and insight to figure out that the question is bogus that to just do the problem straight and get a sensible answer.
We should be encouraging thinkers not robots.
How much are these idiot examiners getting paid. They obviously didn’t proof read the paper at all. No doubt no one will be held accountable.
The history teachers do the proof reading.
Every year there’s a problem with the Higher Maths paper. Now that they’re awarding extra points for the paper, can they not put that little bit of effort into making sure it’s smooth sailing?
Pun intended?
For the second part of the question I combined the two diagrams to create a rhombus (anyone who did the paper will know what I mean!). I was stuck on it for about 10 minutes so when I managed to think of a solution I was beyond delighted. But when it worked out to be a seven figure answer (that was clearly wrong), I was stumped. Hopefully the marking scheme will be sound enough and I’ll get most marks!
Jesus! You did what now!? I’d still be eating Pritt Stik and shaking uncontrollably… Good luck in the morning!
I’m sure they’ll get over the shock eventually. Hopefully before they encounter something that is actually traumatising in the real world.
They’re not in Hogwarts; school is part of the real world. Expecto Patronising!
Actually the question was easier with the additional information as it allowed students to used the Sine rule rather than the Cosine rule. Both rules gave different answers, however, due to the incorrect information being presented. As Lorcán was saying, it’s unlikely that students would go back and check their answers using the more difficult method and could just proceed to further parts of the question with their answer from the first part (which would technically have been correct and examiners could not have penalised further parts of the question for using it). I really genuinely don’t think this is nearly as big a problem as is being made out. Exam mistakes do happen both at Leaving Certificate level and, more frequently in fact, at university level and maths is more prone to them than other subjects. I think Eamonn is also taking advantage of the situation for some free publicity.
As written the question is written it is looking for the length between the point R and the line HP, which is a set of values ending in 110km and starting at half of the length of the line HP divided by the area of the triangle HRP, this is unanswerable as the area is not provided.
I’m a horticultural student, if I can spot that the trig question is suddenly an unanswerable question about sets and geometry then a team of professionals who are being paid to write the exam should have spotted this error too.
David,
I see what you mean, but the distance from a point to a line is understood to mean the perpendicular distance, not the distance in a random direction, so this was not one of the issues. The fact that the area isn’t provided is not relevant to the question either. There was more than enough (in fact, too much) information already provided.
Hope that helps,
Eamonn
@ Mr. Toland. There is no confidence to shake in the first place.
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