Mathematics Teaching in a Second Language in Ireland
Máire Ní Ríordáin
John O’ Donoghue
Department of Mathematics & Statistics,
Department of Mathematics & Statistics,
University of Limerick
University of Limerick
[email protected]
[email protected]
Paper presented at the British Educational Research Association Annual
Conference, University of Warwick, 6-9 September 2006
Abstract
Language plays an important role in the teaching and learning of mathematics. This
investigation is focused on learners (Gaeilgeoirí) in the transition from learning
mathematics through Gaeilge (Irish) to learning mathematics through English.
Currently, there are a significant number of primary and secondary students studying
through the medium of Gaeilge in Ireland, who face an imminent transition to English
medium education, be it at second or third level. This paper presents the findings,
conclusions and recommendations of a preliminary study and exploratory research
carried out by the author in an Irish context. The purpose of the studies respectively
was to investigate the extent to which performance in mathematics could be attributed
to the language of instruction and examine the experiences of Gaeilgeoirí in the
transition process. These studies were carried out at key stages of transition in
education – from primary to secondary and from secondary to third level.
Introduction
Language and communication are essential elements of learning and teaching
mathematics (Gorgorió & Planas, 2001). Therefore the language that we initially learn
mathematics through will provide the mathematical foundations to be built upon and
developed within that language. A characteristic feature of the Irish primary and postprimary school system is that the curriculum can be mediated in either Gaeilge (Irish)
or in English. Teaching and learning through the medium of Gaeilge is the natural
school environment for teachers and pupils in the Gaeltacht areas (Irish speaking
districts) of Ireland. Gaeilge is the mother-tongue of the community and thus it is
natural for children to learn through this medium. “The Irish language as the national
1
language is the first official language.” (Bunreacht na hÉireann, Article 8, 1937).
Since the foundation of the Irish Free State in 1921, the education system has been
recognized and utilized as a basis of the movement for fostering Irish-English
bilingualism (Education Act, 1998, Pr. 1, Section 6).
A parents’ initiated movement in the early seventies was responsible for the
establishment of Gaeilge-medium primary and secondary schools outside of the
Gaeltacht regions, known respectively as Gaelscoileanna and Gaelcholaistí (Coady &
Ó Laoire, 2002). The number of students attending these schools has seen an increase
of more than sixty per cent in the last decade, with the number of schools increasing
by more than fifty per cent (Sunday Independent, March 12, 2006). Currently there
are 143 primary schools (9,556 pupils) and 27 secondary schools (4,841 pupils)
located in Gaeltacht areas (Mac Donnacha et al., 2005). Similarly, 158 Gaelscoileanna
(primary schools) and 36 Gaelcholaistí (second level schools) have been established
with an estimated 31,000 pupils attending these schools (Fás ar an nGaelscolaíocht sa
Ghalltacht, 2005). Using educational statistics from 2004 approximately 6.8% of final
year primary students are learning through the medium of Gaeilge with 2.5% of the
secondary school population studying mathematics through Gaeilge (Tuarascáil
Staitistiúil, 2003/2004). Hence, there are considerable numbers of students
transferring to English medium education annually, be it at second or third level
education.
The authors anticipate significant difficulties arising in Science, Engineering and
Technology (SET) subjects - mathematics in particular, at these key stages of
transition in education, from primary to secondary and from secondary to third level.
Students have the option of transferring to learning through English at second level,
whereas English medium education is the norm at third level. Students submerged in
these transitions will be required not only to learn mathematics, but also to learn
mathematics through English (Barwell, 2003). This paper focuses on issues in these
transitions and discusses relevant research, methodologies and findings from a
preliminary study (Ní Ríordáin, 2005), and from follow-up exploratory research
carried out in the academic year 2005/2006.
Mathematics and Language
2
Associations have been made between learning a language and the learning of
mathematics, since mathematics has a specific register and set of discourses (Setati,
2005). The mathematics register consists of specialist vocabulary (Gibbs & Orton,
1994) and it is the language specific to a particular situation type (Lemke, 1989).
However the mathematics register should not be thought of as only consisting of
terminology. It also contains words, phrases and methods of arguing within a given
situation, conveyed through the use of natural language (Pimm, 1987). Within the
mathematics register different forms of mathematical language can be found (Figure
1.1.). This reinforces the view that the content of mathematics is not taught without
language.
Everyday
language
e.g.,
let’s use the string to
measure how far it is
around the top of the
bucket.
Mathsspecific
e.g.,
calculate the
circumference of the
circle using the formula
APPLIED MATHEMATICS
Symbolic
e.g.,
C = 2πr
Figure 1.1. Diagram showing types of mathematical language (adapted by
Bubb, 1994 from Ballard and Moore, 1987).
More recently criticisms of the mathematics papers in Irish state examinations (June
2006) at both Junior and Leaving Certificate level included reference to the increase
use of "an awful lot of language" as opposed to the use of maths symbols and that this
may be a barrier for those for whom English is not a first language (Irish Examiner,
June 9th, 2006, p.8). This increased use of language was deemed necessary in order to
assist students in achieving a better understanding of the mathematical tasks; however
this strategy is a two-edged sword and may have adverse effects on those not
competent in the language of instruction. The process of learning mathematics
involves the mastery of the mathematics register (Setati, 2005). This allows students
to communicate their mathematical findings in a suitable manner but “… without this
3
fluency, students are restricted in the ways that they can develop or redefine their
mathematical understandings.” (Meaney, 2005, p.129).
Mathematics Learning and Bilingualism
The issue of learning and teaching mathematics in a second language has been studied
extensively in other countries (USA, Canada, South Africa, Australia, New Zealand,
Spain & UK). Currently, two main second language learning contexts have been
identified - submersion and immersion. These programmes lead to subtractive and
additive bilingualism respectively (Bournot-Trites & Tellowitz, 2002). Subtractive
bilingualism usually occurs when a majority language replaces a minority language
(Lambert, 1990). Students are forced to adapt quickly into mainstream education
where the majority language is used as the medium of instruction. This occurs when
students from Gaeltachtaí (Irish districts) transfer to learning through the medium of
English. In the case of additive bilingualism the mother-tongue of the child is the
majority language and they opt to study through a second language, which is a
minority language (Bournot-Trites & Tellowitz, 2002). This is comparable to students
in Gaelscoileanna/Gaelcholaistí who choose to study through the medium of Gaeilge.
There are conflicting views about the learning of mathematics in a second language at
all levels of education. The difference between the two types of bilingualism is
important. Some studies (immersion programmes) have found positive correlations
with learning mathematics in a second language and academic achievement (e.g.
Barwell, 2003; Bournot-Trite & Tellowitz, 2002; Clarkson, 1992; Cummins, 1979;
Swain, 1996; Turnbull et al, 2000, Williams, 2002). While other studies (submersion
programmes) put forward concerns that such pupils underachieve in mathematics (e.g.
Adetula, 1990; Barton et al, 2005; Galligan, 1995; Gorgorió & Planas, 2001; Marsh et
al, 2000; Secada, 1992; Adler & Setati, 2000). The only Irish study in this general
area was carried out by McNamara (1966) and he found that learning through a
second language hinders progress in mathematics. However, the conditions of
McNamara’s (1966) study are different to those identified in this research project – he
was concerned with students with English as their mother-tongue and were studying
through the medium of Gaeilge at primary level, which was compulsory in the chosen
primary schools at this particular period of time.
4
Several studies have demonstrated a link between a student’s proficiency in their first
and/or second language and cognitive processing (Galligan, 2004; Secada, 1992;
Silby, 2000). This is supported by linguistic theories, the most prominent being
Cummins’ Threshold Hypothesis. This theory states that the level of first and second
language competence reached by a student determines if he/she will experience
cognitive deficits or benefits from learning in a second language (Cummins, 1979).
He also distinguishes between basic interpersonal communicative skills (BICS) and
cognitive academic language proficiency (CALP). What is important to note here is
that, while second language learners may pick up oral proficiency (BICS) in their new
language in as little as two years, it may take up to seven years to acquire the
decontextualised language skills (CALP) necessary to function successfully in a
second language classroom (Cummins, 1979). This theory is furthered by his
Interdependence Hypothesis which proposes that the greater the level of academic
language proficiency developed in the first language the stronger the transfer of skills
across to the second language (Cummins, 2000). Although these theories have led to
controversial debates among academics, they have influenced educational policies in
the USA and in the UK (Yushau & Bokhari, 2005).
Issues in Teaching and Learning Mathematics through the Medium of
Gaeilge/Irish - The Preliminary Study
As part of a final year dissertation (2005) the author carried out pilot studies to
investigate the issue of transferring from learning mathematics through Gaeilge to
learning mathematics through English. The aim of the dissertation was to investigate
the extent to which performance in mathematics could be attributed to the language of
instruction and to examine the experiences of Gaeilgeoirí in the transition process.
These studies were carried out at key stages of transition in education – from primary
to secondary and from secondary to third level.
Because of the diverse nature of the topic, a co-ordinating framework that exposed the
various components of the field was needed. Both Ellerton’s (1989) and Gawned’s
(1990) frameworks provided a theoretical structure for the design methodology. These
models demonstrate how language and the language of instruction are key areas in the
learning and teaching of mathematics. They establish the need to link the various
aspects of language factors in mathematics learning such as socio-linguistics, natural
language, psycho-linguistics, problem solving and classroom discourse which
5
intersect with each other. Both qualitative and quantitative methodologies were
employed in the studies due to the diverse nature of the issue being investigated.
These included the use of an adapted Newman Research Method (1977), which
focuses on the written aspects of mathematics and its link with language, and involved
data collection through the use of word problems. These word problems were devised
based on the Heller and Greeno Classification (1978). This classification allows for
easy analysis of and comparison of answers, and, due to only one operation being
required in finding a solution it limits the risk of mechanical operations being a source
of error. A questionnaire was designed by the author in order to assess third level
students’ perceptions of learning through the medium of English, their second
language.
The pilot studies revealed that Gaeilgeoirí under-perform and experience a variety of
difficulties due to the medium of instruction (English). They also highlighted a lack of
resources, teaching strategies and support available to those in the transition from
learning through Gaeilge to learning through English. Recommendations included
further investigation in relation to the particular aspects of the mathematics register
and the English language that hinder Gaeilgeoirí’s learning of mathematics and cause
difficulties for the students learning in a second language. Also it was suggested that
Cummins’ Threshold Hypothesis should be investigated in relation to Gaeilgeoirí in
order to assess whether the level of language competence in both languages (Gaeilge
& English) has detrimental/beneficial influences on their learning.
While the undergraduate study confirmed underperformance by Gaeilgeoirí, it did not
identify the nature of the specific contributing factors, which the current study intends
to investigate. It is important to note that although these students may function
effectively with English in the daily, routine aspects of communication, they may not
deal with English as effectively in the specialised contexts of mathematics, science
and other subject areas (Cummins, 1979). Also, given that there exists, in Ireland,
educational policies promoting bilingualism it should be anticipated that learning
issues might arise (Education Act, 1998).
Exploratory Research
6
The aim of the exploratory research was to establish and clarify the key issues facing
Gaeilgeoirí in the transition from learning mathematics through Gaeilge to learning
mathematics through English. This is to ensure that subsequent research to be carried
out by the author will address the relevant issues and contribute to development in this
research domain.
Methodology:
In choosing an exploratory methodology the primary rationale is that it is the most
appropriate for the context and will fulfil the aims of the research (Goodson & Sikes,
2001). A qualitative approach was employed in this exploratory phase, with the
purpose of eliciting information to help describe and understand the participants’
transition from learning mathematics through Gaeilge to learning mathematics
through English (Leedy & Ormrod, 2001). More specifically, a Mathematics Life
Histories approach was utilized in which the use of semi-structured interviews
provides for the gathering of narrative accounts of the subjects’ experiences (Coben &
Thumpston, 1994). Interviews were conducted with first year mathematics students at
third level, and with professionals who have experienced the transition during their
education/training. In total six subjects were interviewed (3 students & 3
professionals) and all subjects were required to have transferred from learning
mathematics through Gaeilge at second level to learning mathematics through English
at third level. It should be noted that when a life histories approach is employed,
sample sizes tend to be small, non-random and purposive (Goodson & Sikes, 2001).
Data Collection:
Semi-structured interviewing is the principal method of data collection when using a
Mathematics Life Histories approach. Hollway and Jefferson (2000) contend that the
questions used should encourage the interviewee to talk about specific
times/experiences, rather than relying on the use of general questions concerned with
a long period of time in the interviewee’s life. The author devised a list of questions,
which were used as a guide when conducting the interviews. These questions were
adapted if necessary for each interview, based on the information obtained from
previous interviews. The questions employed were constructed using the ‘TAP’
acronym, as formulated by Foddy (1993) i.e. the topic should be clearly defined,
applicable to the respondent and with a specified perspective.
7
The interviews were recorded using an IC Recorder and the one-one conversations
lasted approximately half an hour. After the interview the recording was transferred to
Voice Editor 3 (voice editing PC software). The interviews were then transcribed and
saved as a Word document. This facilitated the importation into NVivo - software
designed for the analysis of qualitative data.
Data Analysis:
NVivo software allows for the creation of nodes, which essentially are labels that
allow for the data to be analysed. Initially eight nodes were created - Mathematics
Experience, Perception of Mathematics, Primary School Experience, Secondary
School Experience, Transfer to Learning through English, Difficulties Experienced,
Learning and Teaching Strategies, Support. These node titles were consistent with the
topics discussed during the interviews. From analysis of the data using these nodes,
four prominent themes emerged, namely Language, Understanding, Personal
Conceptions and Culture. The data was then reanalysed using these as node titles.
Discussion:
From the reanalysis a number of relationships were established between the nodes
(Figure 1.2). These relationships are shown diagrammatically and discussed in pairs in
the following paragraphs.
LANGUAGE
PERSONAL
CONCEPTIONS
CULTURE
UNDERSTANDING
8
Figure 1.2. The Relationship between Nodes.
Language – Personal Conceptions:
Mathematics terminology was the primary source of difficulty when transferring from
learning through Gaeilge to learning through English for all subjects interviewed. In
particular, it was the “basic maths” or fundamentals that they had learnt through
Gaeilge at primary/secondary level. These “basic maths” included simple operations
such as addition (suimiú), subtraction (dealú); types of numbers e.g. integer (uimhir),
complex (casta); and labels e.g. denominator (ainmneoir), hypotenuse (taobhagán).
“I couldn’t understand a word of what the lecturer was saying because I had no idea
of what orthogonal meant because I had never heard the word before. Things like that
and not understanding what she was looking for because I didn’t know the English
words.”
(Interviewee No.3, 11/04/2006).
Also highlighted by the subjects was the “self-explanatory” characteristic of
mathematics words in Gaeilge in comparison to their English equivalent. This was of
immense benefit for the subjects when studying mathematics through Gaeilge as it
facilitated the development of their understanding of mathematics. The structuring of
mathematical problems in English was also a cause of confusion as it varies from the
structure of Gaeilge and thus the subjects found it difficult to “pick out the important
information needed for the question” (Interviewee No. 4, 07/04/2006).
However, given the huge emphasis that all the subjects placed on mathematics
terminology it was surprising that the majority of them did not perceive language as a
component of mathematics. Perceptions of mathematics ranged from “just solving
stuff”, to “loads of formulas for solving problems” to “figures and computations and
working out answers”. However, the interviewees that had pursued mathematics
beyond degree level had a more developed understanding of mathematics and could
see its relationship with everyday life. But yet their personal conceptions of
mathematics failed to develop an association between mathematics and language.
9
“..with mathematics you don’t have a huge element of language, even when you’re
writing it you’re working it out in numbers and symbols, you’re not working it out in
Gaeilge or in English. So the actual physical working out of a problem isn’t done in a
language.”
(Interviewee No. 1, 27/03/2006).
Language – Culture:
One of the fundamental requirements of existence is the ability to participate in a
culture and participate in situations arising from this culture. Appropriate linguistic
choices are made so as to maximise understanding by the listener/reader (Meaney,
2005). English medium education is largely the norm in Irish third level institutions
thus catering for the majority and isolating the minority. The transition from Gaeilge
medium second level education to English medium third level education was
described by the interviewees as “difficult”, “hilarious!”, “hell” and “being thrown in
the deep end”.
One of the subjects expressed the difficulty and discomfort he experienced on
entering third level education due to his language and cultural background.
“I suppose I’m from a very Irish place, Gaeilge is the main language spoken there
and even the culture is different. I’m seen as the ‘culchie’ boy from Galway and even
the way I speak is different.”
(Interviewee No. 6, 03/04/2006).
But it was not just language that caused a barrier. A number of other issues became
prominent such as the culture of third level institutions, which are directed towards
independent and self-directed learning. None of the subjects interviewed had
approached their lecturers/tutors about difficulties they were experiencing due to the
language barrier and few were aware of support services available to them. Only one
of the interviewees had availed of support provided by their mathematics department.
The general consensus was a fear of being perceived as “different (because of their
language background)”, “looking stupid because you don’t know the maths” and a
fear of lack of understanding of the issue on behalf of those providing support. All
10
relied heavily on the support of close friends whom they felt comfortable talking to
about their mathematics problems.
Language – Understanding:
The concept of understanding has been the focus of much research undertaken in the
past and up to recent times. On one side of the spectrum is Piaget’s work and his
theory of the growth of understanding in children – primarily that language is an
instrument for articulating our thoughts (Snowman & Biehler, 2003). Others such as
Vygotsky, Bruner and Skemp take a different perspective on understanding – that
language shapes and is a requirement for thoughts (Yushau & Bokhari, 2005). One
dominant proponent of the latter theory is the Whorf-Sapir hypothesis – that different
languages arrange reality differently for their users (Prins & Ulijn, 1998). It follows
from interpretation of this theory that the language we speak facilitates our thinking
and our perception, and that “concepts not encoded in their language will not be
accessible to them, or will at least prove difficult” (Durkin & Shire, 1991, p.12).
Although conflicting views exist, what is of importance is that language plays a role
in developing our mathematical understanding.
Many types of mathematical understanding have been identified. These include
Skemp’s relational understanding (knowing both what to do and why) and
instrumental understanding (rules without reasons) (1978, p.9). Likewise, Byers and
Herscovics’ (1977) model defined understanding as being intuitive (reaching solutions
without prior analysis) and formal (ability to connect mathematical ideas and form
chains of logical reasoning) (Herscovics, 1996). A more recent development is
Hiebert and Carpenter’s (1992) conceptual and procedural knowledge. Conceptual
knowledge is necessary for understanding the mathematical processes while
procedural knowledge is associated with knowing the sequence of actions.
Discussion of these types of understanding, as well as the relationship between
language and understanding became an importance aspect of the mathematical life
histories.
It
soon
became
apparent
that
the
interviewees
associated
learning/“knowing” their mathematics through Gaeilge with understanding and
11
comprehending their mathematics. Four of the subjects relied heavily on translation of
terminology/mathematical questions in order to help them to complete tutorial sheets
and exam questions during their first year at university. In particular one subject
found that
“…translation helped ease the stress if I didn’t know something. Like if I translated it
back to Gaeilge I would be like ok I know what this is so I can do it.”
(Interviewee No.5, 31/03/2006).
Likewise, having English as the medium of instruction influenced their learning
strategies. All subjects acknowledged that they “learnt stuff off” for examinations and
that they didn’t understand some of the concepts behind the mathematics that they
were learning. They described learning the English mathematics register as
“relearning the words”, not as developing mathematical understandings through
English and transferring skills learnt through Gaeilge to English. A considerable effort
was dedicated towards rote learning and practice of mathematical questions. One
subject described it as
“..learning the rules all over again. They are hard enough to remember as it is
without the added confusion of changing the language.”
(Interviewee No. 2, 28/03/2006).
Thus the subjects relied on procedural knowledge with little development of relational
understanding.
Personal Conceptions – Understanding:
“It’s (maths) ok. If I understand it I enjoy it” (Interviewee No. 3, 11/04/2006). The
subjects’ experiences in mathematics at all levels of education were intermingled with
their perceptions of their “ability” and understanding of mathematics. Three of the
subjects had positive experiences at primary and secondary level and thus opted to
study degree courses with mathematics as a large component. They felt confident in
completing mathematical tasks and felt that the transition to English medium
education didn’t hinder their progress in mathematics long term. The remaining three
12
had negative experiences, particularly at second level, and were pursuing degree
courses where mathematics was only required at service level. One subject in
particular saw mathematics as “..an eight foot wall that I couldn’t get over.”
(Interviewee No. 6, 03/04/2006). Moreover they felt that they lacked ability in
mathematics and this in turn was a source of negative attitudes towards the topic.
Culture – Understanding:
The various types of understanding were discussed in a previous section and when
these are considered from a cultural (and pedagogical) perspective, they are reflective
of that particular culture’s belief about the learning of mathematics (Davidenko,
2000). The majority of the participants spoke of the new environment they were
suddenly immersed in –
“Going from secondary school to college is hard because you’re in a lecture with 300
more students and the lecturers don’t teach as such, they just read off slides.”
(Interviewee No. 2, 28/03/2006).
Subjects spoke of the expectation of having a certain competency in mathematics
before entering university, but that this was hindered by the level of mathematics
taken at second level and also by the language barrier. Abandonment was a significant
issue for three of the subjects in particular. Progressing from having been “spoon-fed”
all through second level, as well having the constant support of teachers, to suddenly
being an insignificant number in a crowd was a difficult transition for them. This is
consistent with universities’ strategy of producing independent learners. However,
where they fail is in providing direction in becoming an independent learner. People
need support and guidance in the right direction. Thus, all of the interviewees relied
on procedural knowledge in order to succeed in examinations, and little time was
given to the development of conceptual understanding.
Conclusion
13
This type of research is in its infancy in Ireland but further more in-depth research is
needed as a matter of urgency considering the increasing numbers of students learning
through the medium of Gaeilge. The majority of these students face an imminent
transition to English medium education, be it at second or third level education.
In this paper we have discussed a preliminary investigation into the learning of
mathematics through a second language in Ireland, concentrating on students making
various transitions in their education. A number of directions for further research can
be seen from the exploratory work. These include:

Further investigation into the mathematics register – in particular the
mathematics terminology that causes confusion in the transition process, and
an examination of the mathematics register in Gaeilge.

An investigation into the effect of cultural background on the transition to
third level education.

Examination of the relationship between language proficiency (Gaeilge and
English) and mathematics understanding.

Investigation of bilingual students’ understanding of mathematics and the
factors influencing the development of their understanding.
From Figure 1.2. it is clear that mathematical understanding is influenced by
language and that the students cultural background and experiences exercise a large
influence in the development of this understanding also.
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