MATHEMATICS from JUNIOR INFANTS to JUNIOR FRESHMAN
Maurice OReilly
It is simply unacceptable that over 16% of ordinary-level students have not achieved a
pass in this subject.
Higher-level maths is traditionally regarded as one of the toughest subjects in the exam,
but based on this year’s marks students have a far higher chance of getting an A grade in
maths than in Irish or English.
There was – and still is – an image problem associated with mathematics.
The first two of these quotations are from the Irish Times of 15th August, 2001: Aileen
O’Donoghue of IBEC and Emmet Oliver, Education Correspondent with the Irish Times
comment on the mathematics results in the Leaving Certificate released that day. The
third, in the context of the reform of a mathematics department by acknowledging
seriously the importance of computing techniques in both teaching and research, is from
Ray Ryan of UCG (now NUIG) in the Society’s Bulletin of March 1990 [1].
Most will agree that the specific issue which Ray Ryan addressed has evolved
substantially during the past decade, yet it appears that his remark has a timeless quality
about it. Mathematics attracts comments in the press and in everyday conversation of
quite a different quality from other disciplines, from revulsion to admiration, from
bewilderment to awe.
In this paper, I will consider some of the recent changes which have taken place in the
delivery of mathematics as a discipline in (the Republic of) Ireland, from the beginnings
of formal exposure to the subject (in Junior Infants, as we call it) to the beginnings of
third level education (in Junior Freshman, as it is known in TCD). This is an absurdly
broad canvass, and I am sure I shall omit many vital elements, yet it seemed like a
sensible idea when Eugene Gath inveigled me to say something about mathematics and
education at this meeting.
The paper begins with descriptions of the elements of the education system emphasising
recent developments and with particular reference to mathematics. Next, issues arising as
a pupil/student encounters transitions in this system are considered. Then the situation is
discussed from some broader perspectives. The paper finishes with some concluding
remarks.
DESCRIPTIONS
The mission of the Department of Education and Science is to ensure the provision of a
comprehensive, cost-effective and accessible education system of the highest quality, as
measured by international standards, which will: (i) enable individuals to develop to
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their full potential as persons and to participate fully as citizens in society and (ii)
contribute to social and economic development. [2]
The opening statement introducing the mathematics curriculum of the Revised Primary
School Curriculum reads as follows. Mathematics may be seen as the science of
magnitude, number, shape, space, and their relationships and also as a universal
language based on symbols and diagrams. It involves the handling (arrangement,
analysis, manipulation and communication) of information, the making of predictions
and the solving of problems through the use of a language that is both concise and
accurate.
Mathematics education provides the child with a wide range of knowledge, skills and
related activities that help him/her to develop an understanding of the physical world and
social interactions. It gives the child a language and a system through which he/she may
analyse, describe and explain a wide range of experiences, make predictions, and solve
problems. Mathematics education fosters creative and aesthetic development, and
enhances the growth of reasoning through the use of investigative techniques in a
mathematical context. It is also concerned with encouraging the child to be confident and
to communicate effectively through the medium of mathematics. [3]
Throughout the eight years of primary school mathematics, six types of skills are
emphasised (applying and problem-solving, communicating and expressing, integrating
and connecting, reasoning, implementing, and understanding and recalling) across five
‘strands’ (number, algebra, shape and space, measures, and data). The introduction to the
curriculum draws attention, inter alia, to mathematics as an intellectual pursuit in its own
right, to the historical and cultural influences that have shaped the subject, and to the
importance of integrating mathematics with all the other subjects taught.
The pedagogical approach advocated in the curriculum is unambiguously constructivist in
the sense that mathematics learning involves the child as an active participant in the
learning process. Existing ideas are used to make sense of new experiences and
situations. Information acquired is interpreted by the learners themselves, who construct
meaning by making links between new and existing knowledge.
The Revised Primary Curriculum is supported in the Department of Education and
Science by the Primary Curriculum Implementation Group and the Primary Curriculum
Support Programme. The work of the recently formed Primary Teachers’ Mathematics
Association (PMTA) provides a forum for teachers, faculty from the colleges of
education and other interested parties to explore and promote good practice in the
classroom.
Second-level education consists of a three-year junior cycle followed by a two- or threeyear senior cycle. The Junior Certificate examination is taken after three years. In senior
cycle there is an optional one-year Transition Year Programme followed by a choice of
three two-year Leaving Certificate programmes [4], namely the Leaving Certificate
General, the Leaving Certificate Vocational Programme (LCVP) and the Leaving
Certificate Applied.
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In the junior cycle (catering for, say, 13-15 year olds), an amended curriculum has been
in effect since September 2000. Significant changes include the introduction of the use of
calculators, a change in the examination format (at foundation and ordinary levels) and a
major in-service programme [5]. About 30% of candidates take higher level mathematics
in the Junior Certificate.
In the senior cycle, the Leaving Certificate General programme has a traditional
academic character and offers curricula in at least thirty subjects including Mathematics
and Applied Mathematics. All subjects – with the exceptions of Mathematics and Gaeilge
– are offered at two levels, ordinary and higher; Mathematics and Gaeilge are offered at
foundation level also. The most recent changes in Mathematics, effected in the early to
mid nineties, introduced the foundation level and made the higher level accessible to a
greater number of candidates. In spite of this, Mathematics is still the subject with the
smallest (by far) participation at higher level. This year, only 18.0% of candidates chose
higher level, 72.5% chose ordinary level and 9.5% chose foundation level. The only other
subjects with less than 50% participation at higher level were: Home Economics
(general) (29.7%), Gaeilge (30.9%) and French (47.5%). As far as Applied Mathematics
is concerned, there are signs that its moribund NCCA Syllabus Committee is about to
stir!
The LCVP, introduced in 1994, aims to balance the virtues of the traditionally academic
Leaving Certificate with the development of skills and qualities which will prove relevant
to the lives of students on leaving school for further education, the world of work, or the
business of making a living [6]. Candidates take at least five (possibly seven) regular
Leaving Certificate subjects together with three ‘link modules’ on Enterprise Education,
Preparation for Work and Work Experience. This year, 20.8% of Leaving Certificate
candidates opted for the LCVP [7].
The Leaving Certificate Applied, introduced in 1995, is a discrete, alternative
programme to the established Leaving Certificate. … The programme is pre-vocational
by nature, aimed at those students who do not wish to proceed directly to third level
education and for those whose aptitudes, needs and intelligence are not fully catered for
by the established Leaving Certificate [6]. In 1999-2000, approximately 8% of senior
cycle students followed the Leaving Certificate Applied.
A glance through the latest IMTA Newsletter [8] bears testimony to the substantial and
varied work of the Irish Mathematics Teachers’ Association throughout second-level
mathematics. It contains articles on: the history of mathematics, geometry, the in-service
support programme for junior cycle, teaching notes, Irish and international mathematics
competitions, the Esat Young Scientist Exhibition, puzzles, book reviews, and solutions
to the Leaving Certificate Applied Mathematics examination (2000). The work of many
colleagues in support of the Irish/International Mathematics Olympiad – well
documented in the Society’s Bulletin – is another example of extracurricular support at
second level.
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In parallel to the junior and senior cycles, the work of the National Council for
Vocational Awards (NCVA) provides an access route to education for those who, for
whatever reason, are outside the mainstream. The NCVA offers awards at four levels –
foundation and levels 1 to 3. No prior certification is required for admission to the
foundation level. Successful completion of one level allows admission to the next. In
addition, admission to levels 1 and 2 can be gained through the Junior and Leaving
Certificates, respectively. The mathematics at foundation level involves basic numeracy.
The material at level 2, for example, is somewhat similar to what is covered at ordinary
level Leaving Certificate, while at level 3, the material is similar to a very traditional first
year business mathematics course at an Institute of Technology. A distinguishing feature
of NCVA courses is an emphasis on coursework (in portfolio or otherwise) – which
might count for 60% of credit – rather than on exams.
Traditionally the third level education system in Ireland has comprised the university
sector, the technical and technological colleges and the colleges of education - all of
which are substantially funded by the State and are autonomous and self-governing. In
addition, in particular in recent years, a number of independent private colleges have
developed, offering a range of mainly business-related courses conferring professional
qualifications, certificates, diplomas and degrees [9]. This sector is presumed well
known to the reader who can find further information from the web-site of the HEA [10]
or, indeed, the Society’s own site.
TRANSITIONS
We have seen that there have been substantial changes in curricula as far as policy,
structure and implementation are concerned during the past decade. The CSO [11]
estimates the population of 5 to 19 year-olds in April 2000 was 893200; a further 326100
were aged between 20 and 24. It is not unreasonable to suppose that the population of
mathematics students is of the order of 750000. However the transitions from one level
(primary, second, third) to the next may not attract sufficient attention. In particular,
dialogue between levels is usually inadequate.
In his article [5], the National Co-ordinator of the Junior Certificate Mathematics Support
Service draws attention to concern expressed by in-service participants to the knock-on
effects for senior cycle mathematics. However no mention is made to the imminent
knock-on effects of implementation of the Revised Primary Curriculum in Mathematics.
The distribution of the 23-volume Revised Primary Curriculum to all primary teachers
and to students in the colleges of education has been a considerable logistical task. It
seems that many others who need to be familiar with how a constructivist mathematics
curriculum is taught, have never seen the two mathematics volumes. Locally, by all
accounts, there seems to be a need to promote much stronger links between primary
‘feeder’ schools and second-level schools about matters of mutual interest and, in
particular, about mathematics.
The opportunities offered by the optional and increasingly popular Transition Year
Programme is realised less in mathematics than in other subjects. It is all too tempting to
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offer pupils a foretaste of, or even an early start to, Leaving Certificate material. Yielding
to this temptation is the norm. On the other hand, innovations using computer packages
such as the Geometer’s Sketch Pad or Live Math swim against this tide. Transition Year
offers a wonderful range of possibilities from modelling problems (such as calibrating a
dip-stick) to investigating how mathematical questions arise from an historical and
epistemological perspective. There are not yet sufficient resources for teachers for
Transition Year mathematics.
Our two opening quotations drew attention to the much-discussed 2001 Leaving
Certificate results of 55144 candidates in mathematics. Participation in mathematics was
higher than any (of 29) other subjects, 6.1% down on 2000, although 3.6% up on 1996.
Let us compare the distribution of mathematics grades with those of Gaeilge.
LC Higher Level
20
2000m
%
15
2001m
10
2000g
2001g
5
0
a1 a1 a2 b1 b2 b3 c1 c2 c3 d1 d2 d3
e
e
e
f
f
f
ng ng
grade
A left skew in the mathematics results at higher level indicates that the best mathematics
candidates are not sufficiently challenged, or, simply, too few are attracted to take this
paper in the first place. A left skew in the results at ordinary level corroborates the
interpretation that many high-performing candidates would be capable of taking higher
level.
5
LC Ordinary Level
%
20
15
2000m
10
2001m
2000g
5
2001g
0
a1 a1 a2 b1 b2 b3 c1 c2 c3 d1 d2 d3
e
e
e
f
f
f
ng ng
grade
The erratic peak at D3 grade (together with the high proportion failing) at ordinary level
indicates that foundation level would be more appropriate for many weaker students. The
Minister for Education and Science has ordered an inquiry into the high failure rate
(16.7%) at this level. It is certainly important to ensure that the national examinations as
assessment instruments are reliable throughout the range of grades. Mathematics is not
the only subject with problems at this level; biology, for example, had a failure rate of
22.6%.
LC Foundation Level
20
2000m
2001m
2000g
2001g
%
15
10
5
0
a1 a1 a2 b1 b2 b3 c1 c2 c3 d1 d2 d3
e
e
e
f
f
f
ng ng
grade
We note that the distribution of grades at foundation level appears normal. It is
unfortunate that participation at foundation level is so low. This may well be due to a
combination of snobbery (in various forms) and the fact that there are many courses at
third level (in particular in the Institutes of Technology) which accept candidates with a
D3 in mathematics at ordinary level, but do not recognise the foundation level regardless
of grade.
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A conference on mathematics in the Institutes of Technology was held in May 1996. The
proceedings [12] are arranged in four parts: students’ mathematical preparedness,
alternative mathematics teaching strategies, computer mediated methods and
mathematics applications research. The first part includes three articles (by Michael
Brennan, John Evans and Seán Ashe) on mathematics in the transition from second to
third level education in Ireland. Dialogue and publication on this issue are rare, and
although this material is five years old, it is still very relevant.
Recently, attention has been drawn to non-completion rates at third level. Performance in
Leaving Certificate mathematics and difficulties encountered by first year students at
third level in mathematics are areas of particular concern. In 1999, a study of first year
students in Institutes of Technology noted: The overall standard of mathematics in the
survey population was quite low: only 8% of first year students had taken Higher Level
Mathematics in the Leaving Certificate compared with 20% nationally in 1996/7. Those
who failed or left were especially likely however to have got a low grade in Leaving
Certificate Mathematics. [13]
Following the HEA study on non-completion [14], one of its authors has since drawn
attention to problems with mathematics: It is striking that in both studies the areas with
mathematical, scientific and technical content are those with the highest level of noncompletion [15]. The studies mentioned draw attention to the importance of developing
strategies to improve student retention, especially in first year.
PERSPECTIVES
Let us now mention some broader and disparate perspectives which have been outside the
scope of our discussion so far.
The use of so-called information and communication technologies (ICT) has had
considerable impact on mathematics teaching at all levels. In 1994, Ted Hurley,
advocating joint courses in mathematics and computer science, wrote [16]: Both subjects
have much to learn from one another. The National Centre for Technology in Education
(NCTE, based in DCU) plays an important role in co-ordinating ICT activities at primary
and second levels [17]. Scoilnet, an initiative of NCTE and NCCA, provides a userfriendly portal for learning using ICT at these two levels [18].
Scoilnet allows users to browse in most curricular areas, and encourages them to
subscribe to any of 262 forums, each under the direction of a ‘facilitator’. Over the past
two years 14692 users made 5628 postings to these forums. Of these, nine forums are
obviously and specifically mathematics related with 826 postings and 10304 readings.
Unfortunately, support for forum facilitators stopped last March. To resuscitate this
service and sharpen Scoilnet in general will take some time.
Proficiency in basic algebra is an area which needs careful attention at all levels. We have
seen that algebra appears explicitly as a strand throughout the Revised Primary
Curriculum. (It appears under the heading, ‘extending patterns’, in infant classes!) At
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third level we know well that serious problems with algebra are commonplace (see, for
example, Michael Brennan in [12]). I have asked many adults who profess difficulty with
mathematics, at what stage they were first aware of these difficulties. By far the most
frequent response was on encountering algebra. It is not clear that significant progress is
being made to diminish the almost total emphasis on mechanical procedures, especially
as far as algebra is concerned in junior cycle. Many pupils are never exposed to the topic
in a manner which they can grasp. This becomes for them a permanent handicap.
Misconceptions around algebra, once established, are difficult to unravel.
Geometry has been in a sorry state in Irish curricula during the past thirty years. As
Paddy Barry has so carefully argued [19], there are wildly different opinions on where
the emphasis should be. Some argue from a pedagogical standpoint, others from one of
axiomatic integrity; some see geometry in terms of a vector space with an inner product,
others in terms of axial symmetry and still others in terms of congruence. In geometry
especially there needs to be co-ordination between textbook authors and the agents of the
Department of Education and Science (the NCCA and the inspectorate). How the
implementation of the new junior cycle curriculum is evolving in this regard remains to
be seen. In any case, I am sure pupils and teachers alike will benefit by exploring Tim
Brophy’s work [20] which brings theorems to life!
For two international perspectives (which are outside the scope of this paper) the reader is
referred to the Third International Mathematics and Science Study [21] and the OECD
Programme for International Student Assessment [22].
REMARKS
In spite of many improvements, it is still the case that too much mathematics teaching
and learning involves the mechanics of the subject without motivation, insight or
understanding. A friend, commenting on his experiences in the sixties, said, “At school, it
was not apparent that there were ideas in maths – like there were in other subjects”.
Mathematics is above all a human activity. Some of us may see it, on occasion, as
Bertrand Russell did: Mathematics, rightly viewed, possesses not only truth, but supreme
beauty – a beauty cold and austere, like that of a sculpture, without appeal to any part of
our weaker nature, without the gorgeous trappings of painting or music, yet sublimely
pure, and capable of a stern perfection such as only the greatest art can show [23]. Yet
mundane human activity outside the classroom impinges on the accessibility of the
subject to the learner. Societal attitudes and changes such as media and peer pressure,
part-time work [24], parental ambition or indifference, all distract attention from the
Platonic ideal of seeking the beautiful truth above all else.
One of the most urgent necessities must surely be the in-service development of teachers
with limited mathematical formation who teach the subject in the junior cycle. Such an
initiative needs to be combined with sharper attention to teaching strategies for
mathematics in Higher Diplomas in Education in our universities.
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Good content is important, but not enough. As new curricula evolve, there is a need to
attend more to implementation, feedback and modifications. Especially an awareness is
required of what the learner has learnt so far. Let us adopt an attitude of looking to the
future, whilst informed from the past.
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ACKNOWLEDGEMENTS
The author expresses his appreciation to the following who contributed to discussions
which supported writing this paper: Seán Ashe, Seamus Bellew, Tim Brophy, Seán
Close, Susan Connolly, Ita Honan, John Maher, Aideen McKevitt, Hugh McManus,
Mark Morgan, Fran Murphy, Cillian O’Reilly, Michael Ryan, Mary Shine-Thompson and
Seán Twomey.
REFERENCES
1. Ryan, R, Use of MACSYMA and MAPLE in Mathematics teaching in UCG, IMS
Bulletin 24 (1990), pp 55-8.
2. Department of Education and Science:
http://www.gov.ie/educ/organisation/21ce33a.htm.
3. Government of Ireland, Primary School Curriculum – Mathematics, Stationery
Office, 1999.
4. Scoilnet: http://www.ncte.ie/scoilnet/ncca/currsupport/postprimary/postprimary.html.
5. English, J, Update on the in-service support programme for the revised Junior
Certificate Mathematics syllabus, IMTA Newsletter 99 (2001), pp 4-5.
6. National Council for Curriculum and Assessment (NCCA):
http://www.ncca.ie/curr.htm.
7. Flynn, S, USI claims bias against vocational Leaving Cert pupils, Irish Times
21/8/2001, p 4.
8. IMTA Newsletter 99 (2001).
9. Department of Education and Science:
http://www.gov.ie/educ/ThirdLeaflet/third%20level.htm.
10. Higher Education Authority: http://www.hea.ie/.
11. Central Statistics Office: http://www.cso.ie/.
12. Cawley, S (ed), A Mathematics Review, Blackhall Publishing, 1997.
13. Healy, M, Carpenter & Lynch, Non-completion in higher education: A study of first
year students in three Institutes of Technology, Carlow Institute of Technology, 1999.
14. Morgan, M, Flanagan & Kellaghan, A Study of Non-Completion in Undergraduate
University Courses, HEA, 2001.
15. Morgan, M, Current Issues and Trends in Course Completion in Ireland: An
Overview, DES/HEA seminar on course completion in Irish higher education,
unpublished, Dublin 28/5/2001.
16. Hurley, TC, Benefits and advantages of an integrated mathematics and computer
science degree, IMS Bulletin 32 (1994), pp 63-72.
17. National Council for Technology in Education: http://www.ncte.ie/.
18. Scoilnet: http://www.ncte.ie/scoilnet/ncca/.
19. Barry, PD, Curriculum development in secondary school mathematics with special
reference to geometry, IMS Bulletin 18 (1987), pp 78-91.
20. Brophy, T: http://www.ncte.ie/tbrophy/.
21. Third International Mathematics and Science Study: http://timss.bc.edu/.
22. OECD Programme for International Student Assessment: http://www.pisa.oecd.org/.
23. Russell, B, Mysticism and Logic, Unwin Paperbacks, 1986 (first published 1917).
24. Morgan, M, How part-time work hit students’ performance, Irish Independent
16/8/2001.
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