This Friday, close to 13,000 students in the Republic of Ireland are set to take higher level maths in the Leaving Certificate, the state exams for 17-18 year old school leavers. That’s the highest number for two decades, and a 25% increase on last year’s all-time low of 10,400 who registered to sit the higher level exams. Typically, only about 80% of those show up for the higher level paper on the day–last year just 8,200 did–the rest playing safe and switching at the last minute to the ordinary level exams.

In 2011, a little over 55,000 Irish students overall, in a country with a population of 4.6 million, sat the Leaving Certificate in their final days of secondary education. This year, just under 54,000 school leavers are taking the Leaving, as it’s known. I hope they’ve studied hard, and wish them every success!

This year’s big increase in those going for the higher level papers is not a random event: for a 4-year trial period, those who achieve a grade D3 or better on the higher level maths exams will get 25 bonus points towards their all-important CAO, which determines what university they get admitted to. It’s all part of a government initiative to reverse a perceived decline in maths performance of the nation’s youth.

It’s also, in part, a throwback to a system that was in place when I took the Leaving Cert many moons ago; in those days we got “double points” (the exact number depending on how well you did) for Honours Maths at a time when there were two levels, *Pass* or *Honours*. You had to pass one of these to enter university. Failing Honours — even though you might have passed Pass had you taken it — would keep you from moving on to third level education. Today, it’s not that different for those who are nervous about their chances, and it explains now, as then, why many of those who plan to sit the Honours or higher level examination papers settle on the day for the safer Pass/General/Ordinary level papers: it’s less risky.

Now, thanks to Project Maths, there are actually three streams/levels, and the situation above has been adjusted a little.

Last year’s Higher Level papers generated some controversy: even before they were unveiled, there were concerns about the decline in numbers already alluded to.

See the papers for yourself: Paper 1 and Paper 2 are available online. Solutions — and plenty of lively student commentary — are also online if you poke around.

Question 9.c on paper two is curiously “self-reflective” (from the point of view of student populations). It starts:

“The mean percentage mark for candidates in the 2010 Leaving Certificate Higher Level Mathematics examination was 67·0%, with a standard deviation of 10·4%. The suggestion that candidates who appealed their results have, on average, similar results to all other candidates, is being investigated….”

It is believed that these papers are the work of one person who has been writing them for a number of years and that, unlike end of year university examination papers in Ireland (and the UK), there are no external examiners who must approve them months ahead of time.

Regardless, a system such as this ensures that one has some idea of what incoming university students know about mathematics. Every single one of them has had 12 years of maths without a break. It simply isn’t optional. And there are no remedial courses such as precalculus, trigonometry or “college algebra” at the third level.

Furthermore, just as in my day, even the pass/general/ordinary level students have to do differential and integral calculus. Here are the 2010 papers to demonstrate that all Irish school leavers know some calculus:

Ordinary Level Paper 1 from 2010

Ordinary Level Paper 2 from 2010 (note Simpson’s rule!)

This is all in stark contrast to the situation in the USA, where I currently live and teach. There, there is essentially no guaranteed minimum level in mathematics that one can expect an incoming university student to have achieved. Some have not taken mathematics for several years before they show up at the gates of third level institutions. As might be expected, that leads to all sorts of issues. But that’s a topic for another day.

Those past papers were real eye-openers! I even looked at the Higher papers from the same year, and it was shocking how short they are, and how low the level of maths being assessed is.

To contrast with England, in June 2010 our students had a choice of

21modules to choose from, of which they had to take six over the course of two years, sitting a 90 minute exam at the end of each one.The specification for the 2011 qualification from AQA was 119 pages long (couldn’t find 2010’s, and we have multiple competing awarding bodies).

I would expect the Higher paper to have some questions corresponding to those in the Further Pure modules, taken by students who do two maths A-Levels, but I can’t see any. Uptake of further maths in England isn’t exactly high, but what are really gifted Irish students supposed to do with themselves?

I think that’s a slightly unfair comment. I do complain that the leaving certificate syllabus has been streamlined since I did it myself some time ago but the requirements for each subject need to be looked at in terms of the structure of the qualification as a whole, which as other comments have pointed out, covers a broader range of subjects than A Levels – I believe it averages out at around 7 for most students and the assessment structure is different. It is worth noting that maths is mandatory for all students at leaving certificate rather than becoming an option post 16. Anyone who is truly gifted at maths would not be stopping at Leaving Certificate level anyway.

For the record, one should not expect the Leaving Cert papers in any subject (or at any level) to compare to A levels in the UK: Irish school leavers typically take final exams in 6-8 subjects in a single three week period in June, so obviously the depth gone into in each subject is limited. As I recall I did 9 subjects, 8 of those at the Honours level. One of those was Applied Maths, which was not taught at my school, but you could teach yourself if you were half decent at maths and physics (the Applied Maths was mostly mechanics).

According to this Wikipedia page, for UCAS (entry to university in the UK), “a Leaving Certificate (higher) subject will be worth two-thirds of an A-level”. Relevant UCAS page is here.

That quote should probably have been “a Leaving Certificate (higher) subject will be worth two-thirds of an A-level [subject]”.

This means six LC HL subjects equally four A-levels.

The results of last Friday’s Irish school leaver high level maths exams aren’t in—for that we must all wait till August—but some comments are.

For instance (from “Higher maths gets mixed response” http://www.irishtimes.com/newspaper/breaking/2012/0611/breaking38.html):

“At first glance it looked difficult, but once you got working on it it was more managable. It was in line with the format we were expecting for the first Project Maths paper.”

“We would have hoped for an easier opener to help settle students nerves about this new style paper. The biggest issue is the poor wording of the questions. Question 4 on probability is an example of this. Also, it is debatable whether or not parts of Q7 on statistics are really on the syllabus.”

and

“Our candidates spent many hours learning about solving trigonomic equations, the sin/cos rule, permutations and combinations, periodic graphs and none of these was touched.”