The phased introduction of the new maths course, and the corresponding exams, continues apace. This new course, called Project Maths, as anybody with the slightest interest in second level education knows, has been the source of much heated debate since it was first introduced a number of years ago.
This year, for the first time, both exam papers will deal exclusively with the new course. So now the old course is history, a distant memory, irrelevant. Nevertheless, for many students, the mindset from the old course remains. As a teacher, I frequently have to field questions from the students in my classes of the type: What exactly is on the course? What is coming up? Can they ask this, that or the other? How many marks will I lose if I leave this out, or make this mistake?
My answers do not always appear to satisfy. I once made the mistake of saying: Don’t worry about the exam, you should concentrate on learning the maths. The reaction of the student who put the question (a super student who would go on to do really well) was one of complete derision and rejection of my reply.
On reflection, I could see where he was coming from. How dare I tell him not to worry about the exam? The exam, and the result he achieves, is his future. It could make the difference between him studying the course with a high points requirement that he really wanted to do or having to settle for something else. Holistic appraisals Like any driven person, he wanted to be given boundaries for what he was expected to do, and within these he would aim to excel. Talking again later, he said he felt a lack of definition and direction. I replied that there was a syllabus that was specific to an extent, but contained many generalisations, and that clarifications I had sought were often not forthcoming, or too vague to be of any material use. However, I felt that he should concentrate on understanding and applying his maths to as many problems as possible.
I also outlined to him the new holistic marking scheme, where the overall performance in a question is judged, rather than a line by line, or symbol by symbol, analysis. I tried to reassure him that the new style marking scheme was very generous, but the caveat was that for a very high grade he needed to be very precise and accurate.
He asked about the reason for the change of syllabus and in particular the change in approach. He has an older brother who studied the old course, loved it, did quite well and is now studying engineering. His brother said that he did not fully understand everything he met in Leaving Cert maths, but because he was “pretty okay” with the skills, he could easily fill in any gaps he had at third level. When he showed this brother the maths he was doing, the reaction was one of disdain.
I pointed out that not all students are like his brother. Many students leave school with no understanding of maths at all, even if they obtain a pass grade in the Leaving Cert. This was a problem that the country as a whole had to face, and that while Project Maths was not perfect, in many ways it was an improvement. More students (not all) should understand more (not everything). In addition, more students will be taking higher level maths than before, and that means more students will have a greater exposure to maths when entering third level. Reasonable objectives One of the objectives of the new course is to increase the number of students taking maths at higher level. This has undoubtedly occurred, but not due to any merits or attractions of the new course. The increase can be squarely attributed to the carrot of 25 extra points on offer to students who achieve at least a D3 grade in higher level maths. This bonus is due to reviewed soon. It is a no-brainer that it will be kept because were it not, the number of students taking higher level maths would plummet.
The original logic was that, with the old course and without any bonus points, many students who could have taken higher level maths didn’t. Instead they chose to take ordinary level and focus on other subjects to accumulate the points for their desired courses. Reasons often given for not taking higher level were the difficulty of the course and, in particular, its length.
Another factor causing the number taking higher level to be so low was the large number of students defecting to ordinary level near to, or even on the day of, the exam. This was largely through fear of failing maths and the catastrophic effect this would have on entry to third level.
It was reasoned that if the top cohort of ordinary level students (those who got As and top Bs) could be persuaded to take higher level maths, then the desired increase in the higher level numbers could be achieved. As there was never any chance that the numbers taking higher level would increase through enthusiasm for the new course, the 25 bonus points were born. Also, the word filtered out from the pilot schools that due to the new method of marking, it would be much harder to fail higher level maths, thus reducing the fear factor.
To a large extent, these actions have had the desired effect. In 2013, just over 13,000 students took higher level maths in the Leaving Cert. This was up from around 8,000 just a few years earlier. The bonus points have put pressure on those students aiming for much sought after courses to take higher level maths, even when their intended courses have little or no mathematical content. This is based on the reasoning that the vast majority of their competitors for places on such courses will go for the 25 point bonus and they cannot justify placing themselves at a disadvantage by not attempting to get those bonus points too. This knowledge also has the effect of reducing the number of late defections to ordinary level maths, although some does still occur.
In addition, the perceived reduction in the numbers failing higher level has, for many students, eliminated the fear factor. However, you should still be careful. Without proper preparation, it is still possible to fail the higher level exam. Under the previous exam, the failure rate at higher level was about four per cent (it was 3.9 per cent in 2007). In 2012, this figure had dropped to 2.3 per cent, but last year it was back up to 3.4 per cent, hardly significantly lower than it was before Project Maths.
There are now more students studying higher level maths and taking it in the exams, and there are more getting at least the D3 grade required for the bonus points. In 2013, approximately 12,570 students achieved a D3 grade or better. When compared to the total number of students taking higher level maths in 2007, which was 8,388, the success of the new course appears obvious. Slipping standards However, there are many implications of this increase in the numbers taking higher level maths. One is the concern about the standard of higher level maths, compared to the old standard and the standard in other countries.
It is not reasonable, I believe, to suggest that before the introduction of Project Maths there were about 4,000 students who were capable of taking higher level maths but didn’t. Nor is it reasonable to propose that the new course, and especially the new teaching method, has improved the maths abilities, so quickly, of this large number of students.
It is clear that the maths standards of the students achieving D grades, and low C grades, has to be considerably lower than the standards of students with comparable grades only a few years ago.
Indeed a number of commentators have expressed concern at the standards of all students, with the possible exception of those achieving A grades. For these A-grade students, who will do well under any course or system, the concerns are different: the fear exists that they will not be adequately prepared for the science, technology, engineering and maths (STEM) courses that many will take. This is due to the change in emphasis in Project Maths from STEM material to material more suited to business and problem solving.
The removal of vectors, matrices and most of integration from the higher level course has been defended by advocates of Project Maths on the basis that a number of third level academics agreed to the change. They declare that the academics’ reasoning was that incoming students did not understand these topics properly and it was preferable to have the material replaced by Euclidean geometry whose principles, supposedly, form the basis for these topics.
I do not doubt that what they say is correct. But I have to wonder, who are these academics? I ask because many other third level STEM academics I have talked to are appalled at the removal of these topics from the second level maths course and feel that incoming students in their areas will be severely under-prepared.
Further evidence of this view can be seen at the Dublin Institute of Advanced Studies, a highly respected academic body set up in 1940. It has begun to offer a summer course in theoretical calculus aimed at students who intend to study mathematics, science and engineering, as well as economics. This would not have happened had there not been major concern with the content of the new course.
In fact, in my experience, corroborated in talks with many teachers, it is not the vectors and calculus that students of the old course did not grasp. It was more probability and inferential statistics that caused problems. Yet these are the very areas that have been expanded, not dropped.
There are other problems associated with the increase in the number of students taking higher level maths. One of these relates to the classroom experience of a higher level maths teacher.
The view from the blackboard If a typical higher maths class had 14 students a few years ago, it has 23 students now. These extra nine students are those who would have taken ordinary level maths in previous years. For the most part, they are the weakest students in the class. From the point of view of the teacher, they require more help and guidance than the other 14. Not that this help is any way begrudged, but it takes a significant amount of extra time: topics have to be revisited more often, extra explanations are often necessary, more incorrect answers are given to questions put to the class, among other issues.
And with all the help in the world, many of these students will still struggle, and often fail, to understand many of the more complicated ideas in higher maths. Yet 12,570 students achieved at least a D3 grade in 2013.
The bottom line is that there is a cohort of students now leaving schools with decent Leaving Cert grades in higher level maths whose mathematical abilities and understanding is demonstrably inferior to the ability of students with similar grades who studied under the old syllabus. The new marking scheme, which most students are now very familiar with, sends out the ambiguous message that it is not necessary to get anywhere near a successful conclusion to a maths question to be awarded at least a “C” grade, if not a “B” grade.
The counter argument that has been made is that the new exam and marking scheme are seeking different abilities than the old course. It is said that what is sought of a student is to demonstrate an understanding of a question, not necessarily to be able to complete all the mechanical steps of a calculation. Yet many third level courses require students to be able to complete calculations and achieve correct answers. The new approach to marking is brewing a storm for many third level courses and students. Brave new syllabus As a teacher, I find much to commend the new course. In many topics, the course content has been improved, in terms of both the understanding of concepts and the use of applications to illustrate the use of the mathematics being learned. The idea of using IT to demonstrate concepts is to be welcomed, and I find that students who react best to visual stimuli relate particularly well to the use of IT in the classroom. I also consider that when the course is fully bedded in, students will have a better understanding of the important concepts and will be more confident using maths in real life situations.
However, there are problems. The course is too long, both in terms of the length and vagueness of the written syllabus, in the demand by students and their parents that all question types be covered, and the increased pressure placed on teachers now working with a wider spectrum of students at higher level.
Anecdotal evidence has it that the response from those in favour of Project Maths, to the claim that the course is too long, is a dismissive rejection and a statement that any teacher who feels that way is not teaching the course properly or as intended. This is arrogant and, more importantly, incorrect. Indeed, if teachers were to teach the course in the manner intended, ie by more investigation, discussion and group work, there is no doubt that the length of time required to complete the course would be significantly increased.
Another suggestion I have heard is that teachers do not need to cover many possible question types in each area. Just explain the concepts to the students, do a couple of quick questions, and after that they can improvise.
This point of view is unsustainable. Students would quickly see from exam papers, talking to friends, etc that they were not being properly prepared and they, or their parents, would demand change. Such an approach would lead to a huge increase in students taking grinds to fill in the perceived gaps in their knowledge. This has already happened in every city and town in the country but it would only get worse.
From talking to many teachers around the country, I know that many of the views expressed here are widely shared. However, I have little faith that such views will be listened to and far less that experienced teachers will be consulted or any action taken.
It is clear, from the reaction of those responsible for bringing in Project Maths to queries and suggestions, that nothing will change, any opposition to the new course is simply opposition to change, and they know best.
No doubt we will all get over it and see the benefits of the new programe, and their wisdom, in due course.
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