Mathematics Teaching in a Second Language in Ireland
Máire Ní Ríordáin
/John O’ Donoghue
Department of Mathematics & Statistics,
University of Limerick /Department of Mathematics & Statistics,
University of Limerick/
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 post-primary 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 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
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.
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 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.
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 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
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.
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.
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.
“..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.”