Polígonos de Thiessen

The concerns, mentioned above, about the standard of mathematics among students leaving second level education, and in particular the standards of those entering third level are identified, by the authors of the NCCA Review, in the Irish context. The Review claims that students proceeding to higher education have a ‘low level of mathematical knowledge and skills’ (NCCA 2005b, 3) and are unable to ‘cope with the basic concepts and skill requirements’ of their courses. This is most probably true in relation to students who achieve the lower grades of the Ordinary Level Leaving Certificate. This is not surprising when one considers that the Chief

Examiner considered that ‘[t]he strengths of Ordinary Level candidates are seen to lie in the area of competent execution of routine procedures in familiar contexts’ (SEC 2000, 31), hence the objective of instrumental understanding is being achieved quite well and is deemed appropriate for young people with this level of competence. Perhaps it is more appropriate to say that these students can apply mathematics in the ‘simplest and most practiced way’ and that this ability is one learned from the routine nature of questions in the Leaving Certificate examination. The authors of the Review admit that the decontextualised nature of the examination questions contributes to the emphasis on routine procedures (NCCA 2005b, 11). It may not be out of place here to suggest that Ordinary Level students, achieving grade D (40%-54%) are, by definition, those whose interest and hence application extends only to routine procedures. These outcomes may differ from those that third level institutions would like to find in their new cohort of students but grade D in Ordinary Level mathematics is the minimum mathematics qualification for many maths related courses, particularly in Institutes of Technology. The problem is in effect an intractable dilemma, since it can only be solved by reducing the number of successful applicants – Institutes of Technology would attract a higher standard of applicant if they raised their minimum requirement, but in turn this would reduce the number of qualifying candidates.

I contend that this perceived problem of falling standards and concern about low level of mathematical knowledge and skill are, in part, connected with the qualifications of students being accepted onto STM courses (as discussed in 3.8) and with the international discourse on mathematics education. The demand for Engineering, Science and ICT courses has been falling for a number of years (Keena 2002, DCU website) and consequently students with fewer CAO points were being accepted onto courses. However, it is beyond the remit of this thesis to fully engage with the question as to whether standards are in fact falling and if so to what extent. Suffice it to say that the NCCA accepted the existence of the problem, based, it seems, almost entirely on anecdotal evidence. In fact the matter of anecdotal evidence and ‘anonymous attribution’ bedevils the entire debate. A proper study of this question would be very beneficial in concentrating the debate.

The Review considers the numbers problem and clearly identifies the role of Higher Level mathematics in terms of entry to third level – the ‘relatively poor take-up of Higher Level mathematics rightly gives cause for concern, since it has implications for the follow-on study of

mathematics to degree level’ (NCCA 2005b, 10). In its discussion of the uptake of Higher, Ordinary and Foundation levels at Junior Certificate it notes that in most subjects Higher Level is intended for the majority of students whereas for mathematics, the Higher Level mathematics syllabus is targeted at students of above average mathematical ability (NCCA 2005b, 8). The result is that the cohort of students who study Higher Level mathematics is smaller than for other subjects. Thus one of the possible causes of the numbers problem becomes apparent – the course is designed for a smaller target cohort. By contrast, the Review attributes the relatively poor up-take of Higher Level at Leaving Certificate in part to the perceived difficulty of mathematics and also to an ‘elitist’ status which, it claims, is attributed to the subject in some schools. No mention is made here of the ‘above average’ cohort although the syllabus states that ‘the Higher course is aimed at the more able students’ (Department of Education 1992b, 5). In keeping with the concerns of business and industry the Review constructs its version of the problem of Ordinary Level at Leaving Certificate. The authors concentrate on the poor performance candidates and construct the problem at this level with what they repeatedly call ‘evidence’. This section reads like a fault-finding exercise rather than a review. Nowhere are the strengths of the system explored. The concentration is entirely on constructing the image of a failed system. Achievements of students are overlooked. The ‘evidence’, predictably now, is selected from international assessments, the Chief Examiner’s Reports, and a healthy dose of anonymous attribution and anecdote from third level institutions. As elsewhere, the role of ‘international studies of achievement’ (NCCA 2005b, 15) in identifying problems with mathematics education in Ireland is accepted and unchallenged. Although the authors caution that the results of such studies must be treated with care as they do not always compare like with like, they still claim that such assessments can provide ‘helpful’ information about strengths and weaknesses in student achievement. They accept the results of international assessments, in which Irish students participated, as evidence that the performance of some students at junior cycle ‘gives cause for concern’ (NCCA 2005b, 16) and that ‘the seeds of at least part of the problem at senior cycle may be sown during the junior cycle or earlier’ (NCCA 2005b, 16). The studies cited are: IAEP I (1998) where the Irish performance was ‘decidedly moderate’; IAEP II (1999) whose results showed ‘a worrying tail’; TIMSS (1994) where the performance of Irish second-year students was ‘better than that of the comparable cohort in a number of countries with similar cultural and developmental level’; and PISA (2000, 2003)

where the Irish achieved a score in mathematical literacy not significantly different from the OECD average but where, in 2003, it was considered that this OECD average performance of Irish students was below that of several countries that might be deemed comparable (NCCA 2005b, 15). This reading of the results already archives the ‘discourse of failure’. The TIMMS 1994, for example, (‘better than that of the comparable cohort’) could have encouraged the authors but they choose to counterbalance the positive message with evidence from the 1996 Chief Examiner’s report for the Junior Certificate which discussed basic weaknesses particularly among students taking Ordinary and Foundation Level papers. In fact, the authors remark that the Chief Examiner’s report ‘helped to counteract any undue optimism from the comparatively good results from TIMMS’ (NCCA 2005b, 16). Equally, the fact that OECD/PISA found the results of Irish students to be ‘not significantly different to the average’ could be read as quite an achievement as the school curriculum is not examined by the assessment and the mismatches are too numerous to mention. In summary, the international assessments and studies, the authors contend, provide ‘evidence that the problems observed at Leaving Certificate level start further down’ (NCCA 2005b, 16). We will return to the subject of international assessment in the chapter 5.

Another ‘evidence’ producing site is in the Chief Examiner’s Report. The ‘evidence’ chosen here points again to trouble with the Ordinary Level Leaving Certificate and also with Ordinary and Foundation levels at Junior Certificate. The authors refer to the reports of the Chief Examiner as valuable documents, which highlight areas of strength and weakness related to the objectives of the syllabus. Predictably the Review concentrates on the weaknesses identified, particularly in 2001 when the ‘extraordinary’ report was produced in response to ‘poor Ordinary Level results’ (NCCA 2005b, 16). A glaring omission in the Review’s consideration of the Chief Examiner’s report (section 5), is its failure to draw attention to the Chief Examiner’s contention that the high failure rate at Ordinary Level is attributable in large part to the failure of third level institutions to accept Foundation Level, a fact that forces students who might otherwise sit that level to take the Ordinary Level paper instead (SEC 2001, 15-16). The issue is briefly dealt with elsewhere in the Review, but at this point the impression of failure is allowed to stand as a general observation.

This Review is an insight into the preferred ‘official view’ of mathematics teaching and learning in this country. It provides an interesting, if limited, overview of certain aspects of mathematics

education in Ireland and contrasts it with some aspects of current international thinking. More significantly, it constructs a discourse of failure surrounding the provision of second level mathematics education, in particular at Ordinary Level Leaving Certificate that would shape its own strategy and that of other actors in the field. The underlying assumptions are clear: that the agenda for the education of young people must be set by international standards and priorities, by international assessments – PISA in particular – by economic considerations and by third level demands; that change is both inevitable and essential. Equally clear is the fact that the agenda for change is already predetermined, that, in fact, the Review is not so much an even- handed analysis of an existing system with all its strengths and failings – the strengths of the present system are ignored – but an unsubtle attempt to construct a problem for which the solution is already known.

The Review discussion document was completed and passed by the NCCA Council in March 2005. At that stage the NCCA commissioned a review of the literature on current international trends in mathematics education as a companion paper to its own document, and to inform the discussion and debate. In the next section we examine the choreography of these reviews, which is interesting in its own right.