The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish

Abstract

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated
through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí)
in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential
element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content
of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics
register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual
students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to
come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key
transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and
applications for mathematics teaching and learning are presented.

Full text

The relationship between performance on mathematical
word problems and language proficiency for students
learning through the medium of Irish
Máire Ní Ríordáin &John O’Donoghue
Published online: 17 September 2008
#Springer Science + Business Media B.V. 2008
Abstract Ireland has two official languages—Gaeilge (Irish) and English. Similarly,
primary- and second-level education can be mediated through the medium of Gaeilge or
through the medium of English. This research is primarily focused on students (Gaeilgeoirí)
in the transition from Gaeilge-medium mathematics education to English-medium
mathematics education. Language is an essential element of learning, of thinking, of
understanding and of communicating and is essential for mathematics learning. The content
of mathematics is not taught without language and educational objectives advocate the
development of fluency in the mathematics register. The theoretical framework underpin-
ning the research design is Cummins’(1976). Thresholds Hypothesis. This hypothesis
infers that there might be a threshold level of language proficiency that bilingual students
must achieve both in order to avoid cognitive deficits and to allow the potential benefits of
being bilingual to come to the fore. The findings emerging from this study provide strong
support for Cummins’Thresholds Hypothesis at the key transitions—primary- to second-
level and second-level to third-level mathematics education—in Ireland. Some implications
and applications for mathematics teaching and learning are presented.
Keywords Bilingualism .Cummins’thresholds hypothesis .Educational transitions .
Mathematics word problems and language proficiency
1 Introduction
Language plays a key role in the teaching, learning, understanding and communication of
mathematics. Mathematics is made meaningful through the use of language and students
should be enabled to communicate adequately in the language of mathematics (Capps &
Educ Stud Math (2009) 71:43–64
DOI 10.1007/s10649-008-9158-9
M. Ní Ríordáin (*):J. O’Donoghue
Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland
e-mail: Maire.NiRiordain@ul.ie
J. O’Donoghue
e-mail: John.ODonoghue@ul.ie

Pickreign 1993). The language we initially learn mathematics through will provide the
foundations to be built upon and developed within that language. A characteristic feature of
the Irish primary and post-primary system is that the curriculum can be mediated in either
Gaeilge
1
(Irish) or English. Since the foundation of the Irish Free State (1921), the
education system has been utilised as a basis of the movement for fostering Gaeilge–
English bilingualism (Department of Education & Science: Education Act 1998, Pr1, 6).
Language is the channel of communication within a mathematics classroom and provides
the tool for student–teacher interaction (Smith & Ennis 1961). Thus competence in the
language of communication/presentation facilitates engagement in the learning process.
However, what is of concern to the authors is the effect that a change in the language of
instruction/presentation has on Gaeilgeoirí’s
2
(students who learn through the medium of
Gaeilge) performance on mathematical word problems.
2 The Irish context
Education has always been highly valued in Ireland. A three-tiered education system has
been established where primary education lasts for 8 years for children between the ages of
four and twelve. The second-level school span is predominantly a 6-year cycle, taken by
ages 12 to 18. Third-level education is provided mainly by universities, institutes of
technology and colleges of education. In the context of primary- and second-level
education, two language options exist as mediums of instruction of the curricula, namely
Gaeilge and English. For children growing up in a Gaeltacht
3
(Irish speaking area), Gaeilge
is the dominant language of the community and the natural language through which
communication and socialisation takes place. Gaeilge tends to be the prevailing language
spoken in the home of these students. In turn, Gaeilge is the medium of instruction
employed in the local primary and second-level schools. Instruction through the medium of
English does not occur and mathematics textbooks are available through the medium of
Gaeilge. This heritage language is held in high regard both by the members of the
communities and by the teachers in Gaeltacht schools. The Gaeltacht areas are revered as
the primary agency for maintaining the Gaeilge language in Irish society. Although a
relatively new phenomenon (mid-seventies), immersion education also exists in Ireland in
which Gaeilge-medium schools have been established outside of Gaeltacht areas. These
schools are known as Gaelscoileanna
4
(primary-level schools) and Gaelchólaistí
5
(second-
level schools) and can be found in all counties throughout the Republic of Ireland. Students
attending these schools are predominantly from English-speaking households and the
communities in which the schools are located are English-speaking-dominated. Parents of
pupils attending these schools view Gaeilge as an important language and the primary aim
1
Gaeilge [Gale-ga]-The first official language of Ireland. More commonly known as Irish.
2
Gaeilgeoirí [Gale-gor-ee]—students who learn through the medium of Gaeilge at primary- and second-level
education.
3
Gaeltacht [Gale-tuck]—District/area in which Gaeilge is the dominant language of the community and the
mother tongue of the children growing up in these areas. There are seven Gaeltachts in total in Ireland.
4
Gaelscoileanna [Gale-skull-in-a]—Gaeilge medium primary level schools located outside of the Gaeltacht
areas.
5
Gaelchólaistí [Gale-coll-awe-stee]—Gaeilge medium second level schools located outside of the Gaeltacht
areas.
44 M. Ní Ríordáin, J. O’Donoghue

in enrolling their children in immersion education is to develop bilingualism. The policy of
immersion education is that all instruction takes place through the medium of Gaeilge.
Teachers employed in Gaeltacht schools and immersion schools tend to originate from
Gaeltacht areas and/or have learnt through the medium of Gaeilge at primary- and second-
level education. Colleges of education do not provide a formal qualification for teaching
through the medium of Gaeilge and this is not a prerequisite for employment in Gaeilge-
medium schools.
The number of students enrolled in Gaeltacht primary- and second-level schools has
remained steadfast over the past decade. However, the rise in popularity of immersion
education is significant and has seen an increase in excess of 60% over the past decade (Fás
ar an nGaelscolaíocht sa Ghalltacht 2005). By combining both the number of Gaeltacht
students with immersion students, it unveils a significant and increasing minority of our
primary- and second-level schools’population learning through the medium of Gaeilge;
approximately 48,000 students in total (MacDonnacha, Ní Chualáin, Ní Shéaghdha & Ní
Mhainín 2005). This equates to 7% of the total primary-level population and 2.5% of the
second-level population. What is of importance to the authors is that the majority of these
students will face an impending transition to English-medium education, either at the
second or third level and, thus, the research undertaken is concerned with the numbers of
Gaeilgeoirí in their final year of primary- or second-level education. Current education
statistics reveal 6.9% of final-year primary-level students are learning through the medium
of Gaeilge, while at the second level, 1.5% of students sit their final examinations through
the medium of Gaeilge. Although Gaeilgeoirí 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 (i.e. subjects with
specific registers) at both key transitions. They will be confronted not only with learning
new mathematics but also with the task of learning it and understanding it through the
medium of English (Barwell 2003).
3 Theoretical framework
Research has demonstrated that language is related to thinking, learning and cognitive
development (Stubbs 1976). Misconceptions about how the brain stores language have led
to negative perceptions of bilingualism, the most prominent being that bilingualism may
result in “cognitive overload”and thus disadvantage the learner (May, Hill & Tiakiwai
2004). This narrow perception of the mind and its storage of language is described as the
Separate Underlying Proficiency (SUP) model, which views the two languages being stored
independently of one another (Baker 2001). Consequently, an increase in one language will
result in an imbalance and loss of a portion of the other language. However, this model is
not an accurate reflection of the working mind. The Common Underlying Proficiency
(CUP) model is a more apt description of language construction within the mind.
Outwardly, both languages are different in conversation. However, internally both
languages are merged so that they do not function independently of one another (Baker
2001). Storage of both languages occurs together and this acts as a central processing unit
that both languages contribute to, access and use (Baker & Prys Jones 1998). Therefore,
given that both languages are dependent on one another, consideration of this needs to be
taken into account when investigating Irish bilinguals and their learning of and
understanding of mathematics. One cannot investigate one language without examining
the other language also.
Mathematics through the medium of Irish 45

A second misconception lies in the belief that many bilingual students appear to
experience restricted educational success, with bilingualism the attribute of liability.
However, research has demonstrated that there are cognitive advantages to be reaped from
being bilingual, which is largely determined by the level of proficiency that a student is
allowed to attain in both their languages. One such theory that provides a framework for
this type of investigation in mathematics education is the Thresholds Hypothesis by
Cummins (1976). His theory states that the level of first- and second-language proficiency
reached by a student determines if he/she will experience cognitive deficits or benefits from
learning in a second language (Cummins 1976). At the first level of this Hypothesis, the
bilingual child has a low level of proficiency in both of the languages and there will be
negative cognitive effects for the student’s learning in mathematics (Baker 2001). At the
middle level, the bilingual child will have age-appropriate proficiency in one of their
languages (comparable to a monolingual child) but not in both. This dominance in one of
the languages is unlikely to influence cognition in any significant positive or negative way
(Baker 2001). The third or top level of this Hypothesis encompasses well-developed
bilingual students who have age-appropriate proficiency in both languages and are likely to
demonstrate cognitive advantages over monolingual or weaker bilingual students in
mathematics (Baker 2001). Given that both languages are interdependent and proficiency
in both is of importance for cognitive performance, the languages cannot be looked at in
isolation as suggested by the SUP model. Clearly, the CUP model is consonant with
Cummins’Thresholds Hypothesis (1976), which reflects the realities of bilingual contexts
while being supported by empirical research (e.g. Dawe 1983; Clarkson 1992).
Although the Thresholds Hypothesis appears to vindicate the dissimilar findings in
bilingual education, there are a number of weaknesses that need to be addressed. The most
prominent criticism relates to the terms used to describe the various bilingual proficiency
levels within the theory which include ‘dominant’and ‘balanced’bilingualism. It has also
been argued that the use of these terms reflects a narrow view of language competence
(Romaine 1989), and accordingly a stagnant perception of language and of the variation of
language use. However, Cummins (2000) has defended these terms as being reflective of
educational contexts, e.g. schools that employ two languages of instruction, and that these
contexts influence the development of bilingualism, e.g. one language may be used more
than the other thus resulting in ‘dominant’bilingualism in one of the languages. Other
criticisms of the model include the vagueness surrounding the proficiency levels at each
threshold and what level of proficiency is necessary in order to avoid negative effects and
facilitate cognitive advantages (Baker 2001). In this study, clear threshold levels were
identified for both Gaeilge- and English-language proficiency at each transition in Irish
education, akin to the method employed by Clarkson (2007). Similarly Hoffman (1991)
questions how one can measure and define ‘educational success’, and suggests that reliance
on traditional measured school tests neglects factors such as motivation, attitudes, social
issues, schooling, parental support, etc. which are important when determining educational
success. Clearly, the issues raised are concerned with terminology and lack of detail but
significant studies have been undertaken that provide strong support for the Thresholds
Hypothesis (e.g. Bialystok 1988, Clarkson 1992; Dawe 1983; Lasagabaster 1998; Mohanty
1994). These studies provide an explanation of the variation amongst bilingual students and
although the theory is controversial in nature, it has influenced educational policies in the
USA and in the UK (Yushau & Bokhari 2005).
Modification of the Thresholds Hypothesis in 1979 looked more closely at the
relationship between a bilingual’s two languages and resulted in the Developmental
46 M. Ní Ríordáin, J. O’Donoghue

Interdependence Hypothesis (Cummins 1979a). This Hypothesis proposes that the greater
the level of academic language proficiency developed in the first language the stronger the
transfer of skills across to the new language in which learning is taking place (Cummins
2000). Conversely, the less developed a student’s first language is, the more difficult it is
to attain bilingualism (Baker 2001). Length of time required in acquiring proficiency is of
importance to this theory. In light of this, Cummins (1979a) 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 2 years, it
may take up to 7 years to acquire the decontextualised language skills (CALP) necessary
to function successfully in a second-language classroom (Cummins 1979a). Mathematics
is located within this CALP and in order for Gaeilgeoirí to attain mathematical academic
language proficiency, their CUP must be well developed (Cummins 1979b). This
underlying ability, in turn, can be advanced through the Developmental Interdependence
Hypothesis and, depending on the type of schooling, either through a student’s first
(Gaeltacht schools) or second language (Gaelscoileanna/Gaelchólaistí). Once again, there
are a number of criticisms of the distinction between language registers, in particular, that
the differentiation underestimates the demands of conversational proficiency, while
overemphasizing the demands of academic proficiency. Also, a potential deficit may be
associated with students who do not acquire academic proficiency (Fredrickson & Cline
1996). However, this model has been shown to have good explanatory power of bilingual
students’relative success/failure when encountering a new language of instruction in
educational contexts.
4 The relationship between mathematics learning and language
Language and communication are essential elements of teaching and learning mathematics,
and this is evident from research carried out in bi/multilingual settings (Gorgorió & Planas
2001). Mathematics itself is a type of formal language. The mathematics register is more
than just vocabulary and technical terms. It also contains words, phrases and methods of
arguing within a given situation (Pimm 1987). This register is conveyed through the use of
natural language and each language has its own mathematics register. Mathematics is not
‘language free’and due to its particular vocabulary, syntax and discourse it can cause
problems for students learning it in a second language (Barton & Neville-Barton 2003).
There are conflicting views about the learning of mathematics in a second language at all
levels of education. Some studies (immersion
6
programmes) have found positive
correlations with learning mathematics in a second language and academic achievement
(e.g. Barwell 2003; Bournot-Trites & Tellowitz 2002; Clarkson 1992; Cummins 1979a;
Swain 1996; Turnbull, Hart & Lapkin 2000, Williams 2002). On the other hand,
submersion
7
programmes have demonstrated that bilingual students underachieve in
mathematics when the school language is different from their home language (e.g. Adetula
6
Immersion Education—Students opt to learn through the medium of a second language with the aim of
developing bilingualism.
7
Submersion Education—Schools/Institutions that contain bilingual students of a minority language, who
are required to learn through the majority language.
Mathematics through the medium of Irish 47

1990; Adler & Setati 2000; Barton, Chan, King, Neville-Barton & Sneddon 2005; Galligan
1995; Gorgorió & Planas 2001; Marsh, Hau & Kong 2000; Secada 1992).
More specifically, empirical studies investigating the relationship between language
proficiency and mathematics performance have been instrumental in furthering this area of
research. Prior to the early seventies, it was assumed that bilingualism had a negative
impact on cognitive development and mathematical learning (Clarkson 2007). Research
investigating the cognitive effect of bilingualism on mathematical learning began in the
early eighties and has progressed from there. In particular the work of Dawe (1983) and
Clarkson (1992) was significant, with Cummins’(1976) framework forming the theoretical
basis of their research. Both Dawe (1983) and Clarkson (1992) concluded that bilingual
mathematics students proficient in both their languages performed significantly better in
mathematics than bilingual students dominant in only one language, and better than their
monolingual peers. They also found that mathematics students who were weak in both their
languages performed poorly mathematically also. This research substantiates the theoretical
idea of threshold levels of language proficiency and this is further supported by research
carried out by Secada (1992) with bilinguals in America. More recent research carried out
at second and third-level education in New Zealand (Barton et al. 2005) with students for
whom English is a second language concluded that these students experience a
disadvantage of between 10% and 15% in mathematics as a result of language difficulties,
which again reinforces the notion of the necessity of language proficiency in both
languages.
Clearly, Cummins’work is renowned and has been confirmed by a large body of
research undertaken worldwide. One may question the significance of undertaking a
similar study and why established findings are not sufficient to inform educational
practice and policy within Ireland. The research undertaken by the authors provides an
account of research on bilingualism and mathematics learning in a new environment
(Irish) and is designed so as to build on previous research on Cummins’Hypotheses while
suggesting some productive lines of further enquiry (see Sections 6&7). As stated by
Ellerton and Clarkson (1996) it is very difficult to compare and generalise findings from
one country to another due to differences in curricula, pedagogies, age cohorts, language
development, cognitive abilities, social backgrounds, etc. Mathematics education is
culturally dependent and specific to the environment in which it is taking place. Thus, it
is necessary for each country to undertake relevant research in relation to bilingualism and
mathematics learning appropriate to the educational context in operation. Ireland possesses
both Maintenance Heritage Language and Immersion Education (Baker & Prys Jones
1998) and these are both firmly established. It thus provides the opportunity for diverse
areas of investigation while contributing to international findings in the domain of
mathematics education.
The research undertaken specifically sets out to examine the influence of language
proficiency on performance on mathematical word problems for Gaeilgeoirí in the
transition to English-medium mathematics education at second and third-level education
in Ireland. The aim of the research includes investigation of the following questions:
&Is the relationship between performance on mathematical word problems and Gaeilge
language proficiency of significance for Gaeilgeoirí?
&Is the relationship between performance on mathematical word problems and English-
language proficiency of significance for Gaeilgeoirí?
&Are Cummins’Thresholds and Developmental Interdependence Hypotheses supported
by an Irish bilingual mathematics context?
48 M. Ní Ríordáin, J. O’Donoghue

5 Performance on mathematical word problems and language proficiency
5.1 Subjects involved in the study
Initially, all Gaeilge-medium primary-level schools in Ireland were identified and contacted
(June, 2006) in order to source potential final-year students in the transition to English-
medium second-level education at the start of the following school year. The school
principals provided names of the second-level schools that these students would be attending
in September. Consequently these second-level schools were contacted via letter and follow-
up telephone calls and were provided with the relevant information on the research project.
Five schools in total agreed to take part in the study. First-year students from all-English-
medium education were also sourced at these schools and were in the same class groups as the
bilingual students so as to facilitate the formation of a monolingual group for comparison.
The bilingual participants at second level were chosen using the following criteria:
&They were required to have studied mathematics entirely through the medium of
Gaeilge at primary level,
&That they were currently studying mathematics through the medium of English at
second level,
&All subjects were in their first year of second-level education.
Both subjects from Gaeltacht schools (16 students in total) and Gaelscoileanna (21
students in total) were used in the study, as well as a control group consisting of
monolingual English-speaking students (49 students in total).
At the transition from second- to third-level education, all Heads of Departments of
Mathematics in universities, institutes of technology and colleges of education were
contacted via letter, email and follow-up telephone calls in order to source Gaeilgeoirí. Only
four institutions agreed to participate in the study, two did not have students appropriate for
the study and the remainder never responded to any of the mediums of communications.
Monolingual students were sourced in each of the class groups that the bilingual students
were located in and were matched according to Leaving Certificate (final examination at
second-level education) mathematics result and overall points achieved. The bilingual
subjects were selected if:
&They had studied mathematics entirely through the medium of Gaeilge at primary and at
second-level education,
&They were now studying mathematics through the medium of English at third level,
&They were in their first year of third-level education.
Once again, subjects from Gaeltacht schools (nine students in total) and Gaelchólaistí
(six students in total) participated in the study, as well as a monolingual control group
consisting of six students who had learnt mathematics entirely through the medium of
English at primary and second-level education. The students selected were from
universities, institutes of technology and colleges of education. Mathematics was a minor
part of their degree courses for all students involved in the study (Table 1).
5.2 Test instruments
At the transition from primary- to second-level education the participants completed a
mathematics word problem test in Gaeilge (bilingual students only) and in English (see
Mathematics through the medium of Irish 49

Appendix Afor a selection of questions used), and language proficiency tests in English
and in Gaeilge (bilingual students only). Word problems can be effectively used in
investigating language issues for mathematics learners in a second language (see Newman
1977) and this is the essence of what we are concerned with in this study. Whereas
mathematics word problems have traditionally been used to determine learners’conceptual
understanding, this study is concerned with the influence of linguistic proficiency in two
different languages on bilingual students’performance on mathematical word problems.
The English mathematics word problem test consisted of twelve questions, with a number
of subparts in some of the questions. A parallel version of the test instrument was
constructed in Gaeilge so that the translation in each language was as accurate as possible
while maintaining appropriate wording in each language (Evans 2007). The word problems
were constructed using standard mathematics textbooks (available in English and in
Gaeilge) for first-year second-level students in Ireland and appropriate piloting took place
in which the students completed the test instruments and teachers provided feedback via a
questionnaire on the word problems utilised so as to minimise difficulty with word
difficulty, content and format in the final versions (Allalouf, Hambleton & Sireci 1999).
Given that the tests had the same content but a different language of presentation the order
in which the tests were administered was changed for every second student in each group.
Thus, half the students completed the word problem test through the medium of English
first and then through Gaeilge, and this was reversed for the remainder of the participants.
This is to ensure the process checked the order effect (Adetula 1990) and there was no
apparent difference in test scores related to the order in which the tests were taken. The
English proficiency test consisted of a standard cloze test available for administration to all
final-year primary school students in Ireland (Wall & Burke 2001). Given that the
participants had just transferred from primary- to second-level education, this proficiency
test was deemed appropriate for their expected level of English-language proficiency.
Currently, no standard proficiency test in Gaeilge exists in Ireland. However, Aonad na
Gaeilge at the University of Limerick has designed an internal proficiency test in
accordance with the Council of Europe’s Common European Framework of Reference for
Language (CEF). The proficiency test provided by Aonad na Gaeilge consisted of 65
multiple-choice cloze questions. However, for the purpose of assessing first-year, second-
level students’proficiency in Gaeilge, only thirty of the cloze questions were used, as this
was the expected level of language proficiency for this age group as advised by Aonad na
Gaeilge. The cloze procedure has been used for a multitude of language purposes (Oller
1975) but it was chosen for this study as it reflects a students’general and specific reading
comprehension ability (Jongsma 1971), a key skill required in solving mathematics word
problems. Thus, a relationship between performance on mathematical word problems and
language proficiency can be established for Gaeilgeoirí in this study.
Table 1 Description of participants at each transition in the investigation
Bilingual group Monolingual group
(English control group)
Total cohort
Primary–second level
(Transition 1)
Entire group (BG–T1): n=37 n=49 (M–T1) n=86 (T–T1)
Gaelscoil (BGc–T1): n=21
Gaeltacht (BGt–T1): n=16
Second–third level
(Transition 2)
Entire group (BG–T2): n=15 n=6 (M–T2) n=21 (T–T2)
Gaelchólaiste (BGc–T2): n=6
Gaeltacht (BGt–T2): n=9
50 M. Ní Ríordáin, J. O’Donoghue

At the transition from second- to third-level education, the participants completed a
mathematics word problem test in English (see Appendix Bfor a selection of questions
used), and language proficiency tests in English and in Gaeilge (bilingual students only).
The English mathematics word problem test consisted of 19 word problems, with a number
of subparts in some of the questions. Appropriate piloting of all test instruments took place.
Sixteen of the word problems were constructed using the PISA mathematical literacy
framework (OECD 2006), which is fitting with our study, and some of the PISA questions
available to the public domain were utilised in the test instrument. The remaining three
questions on the mathematics word problem test consisted of cloze-type questions (see
Hater & Kane 1975). The questions involved definitions or explanations of mathematical
terminology employed in a regular mathematics lecture/tutorial. A standard English cloze
proficiency test was sourced through the Cambridge Certificate of Proficiency in English
(Cambridge Examinations Publishing 2002). The Gaeilge proficiency test provided by
Aonad na Gaeilge was utilised. All questions were included for the second- to third-level
transition as the proficiency test is designed for people aged 18 onwards.
5.3 Analysis
All the data collected were coded and imported into SPSS (version 13) for quantitative
analysis. A technique devised by Clarkson (2007) was used to segregate the participants
into language proficiency groups. In accordance with their score on the language
proficiency test in English, the participants were selected as having comparatively high,
middle or low proficiency in English. By rank-ordering the scores obtained by the
monolingual English control groups, the two scores that divided each group into thirds were
recorded and then applied to the bilingual groups, resulting in three sub-groups at each
transition. The median score for the proficiency test in Gaeilge was used in order to divide
Gaeilgeoirí into comparatively high or low proficiency groups in Gaeilge, at each transition
(Clarkson 2007).
Students were then categorised as relatively high proficiency in both languages;
dominance in one language (combination of high/low); or relatively low proficiency in
both languages (combination of low/low). Each student was assigned to only one of these
language proficiency groups. Six of the students dropped out of the analysis at the
primary–second-level interface because they did not fit clearly within the sub-categories
due to having a combination of high/medium or low/medium proficiency in the languages.
At the third level, two of the students were not included in the analysis as once again they
did not fit clearly within the designated categories (Table 2). The relevant variables in each
of the data sets were explored and tested for normality before applying Pearson’s
correlation test. Significance of the relationships explored was 0.05 or less for the results
reported in Section 5.4.
Table 2 Description of the language proficiency groups
Categorisation Primary–second level (n) Second–third level (N)
High/High High Gaeilge & high English 14 3
Low/Low Low Gaeilge & low English 7 3
Dominant Gaeilge High Gaeilge & low English 10 4
Dominant English Low Gaeilge & high English 0 3
Monolingual All-English schooling 49 6
Mathematics through the medium of Irish 51

5.4 Results
The first concept explored is the relationship between performance on English
mathematical word problems and English-language proficiency (Table 3). When taking
the entire group of participants at the primary- to second-level transition, it was found that
performance on mathematical word problems and language proficiency in English was
moderately correlated (r=0.48). When looking at the two individual groups within this
cohort similar findings were evident with both the monolingual group and bilingual group
displaying modest but significant correlations between performance on mathematical word
problems and English-language proficiency (r= 0.52 and 0.41, respectively). At the
transition to third-level education, a stronger correlation (r=0.69) between performance
on mathematical word problems and language proficiency in English was evident for the
entire group. In particular, for the monolingual English group, performance on
mathematical word problems and English-language proficiency was highly correlated at
r=0.91, while a strong relationship was also evident for Gaeilgeoirí (r=0.65). Therefore, it
is apparent that performance on mathematical word problems in English is related to
language proficiency in English for Gaeilgeoirí and for monolingual English students at
both transitions, with greater importance at the transition to third-level education.
Further analysis investigated the relationship between performance on mathematical
word problems (in English) and language proficiency in Gaeilge for Gaeilgeoirí (see
Table 4). This was particularly significant at the primary- to second-level transition where a
strong relationship was evident for the all Gaeilgeoirí (r=0.65). This group of Gaeilgeoirí
can be segregated further in relation to the school type attended, i.e. either a Gaeltacht
school (BGt) or a Gaelscoil (BGc). For the Gaeltacht group, performance on mathematical
word problems in English was strongly related to Gaeilge language proficiency with
Pearson’s correlation equal to 0.71. For the Gaelscoil group, a moderate relationship is also
evident (r=0.62). However, these findings were not replicated at the transition to third level
where moderate relationships were found not to be significant for either of the groups.
Also, Gaeilgeoirí at second level completed a mathematics word problem test in Gaeilge
and performance in this test is moderately correlated with the students’proficiency in
Gaeilge (r=0.55). Gaeilgeoirí’s performance in the mathematics word problem test in
English was highly correlated and significant with their mathematics performance in the test
through Gaeilge (r=0.81). Clearly, there is a strong relationship between Gaeilgeoirí’s
Table 3 Correlations between performance on mathematical word problems (in English) and English-
language proficiency*
Pearson’s
correlation
Significance (p) Description
Primary–second level T–T1: r=0.48 <0.01 All are moderate correlations but
are significant
M–T1: r=0.52 <0.01
BG–T1: r=0.41 <0.05
Second–third level T–T2: r=0.69 <0.01 Moderate correlation
M–T2: r=0.91 <0.01 Very Strong correlation
BG–T2: r=0.65 <0.01 Moderate correlation
All correlations are highly significant.
*T=Total Group: inclusive of bilingual and monolingual students
M=Monolingual students. BG=Bilingual Group
52 M. Ní Ríordáin, J. O’Donoghue

performance on mathematical word problems in English and in Gaeilge and their
proficiency in the Gaeilge language at the transition to second-level education. Overall
there was a difference of 8.7% in performance between the English and Gaeilge maths word
problem test, with Gaeilgeoirí performing better in the Gaeilge version. This finding has
significant implications as it suggests that Gaeilgeoirí may not be achieving their maximum
potential in mathematics when assessment is through the medium of English. Also, given
that Gaeilgeoirí at this transition stage (primary–second-level education), on average,
perform better on mathematical word problems than their monolingual peers through the
medium of English, the difference in performance between bilingual and monolingual
students may be more significant if language is taken into consideration.
The authors would like to draw attention to some important considerations in relation to
the findings at this stage. No pre-testing took place at primary and second-level Gaeilge-
medium education before the students entered English-medium second-level or third-level
mathematics education. The authors had no contact with these students until they entered
English-medium education. The test instrument administered at second level was designed so
that all material had been taught through the medium of English at second level and had been
completed by all students who participated in the study through the medium of English at
second-level education. The authors concede that some of the material may have been learned
previously through the medium of Gaeilge at the primary level but we can not assume this for all
Gaeilgeoirí, nor can we assume particular questions were covered by all Gaeilgeoirí through
the medium of Gaeilge previously before entering English-medium second-level mathematics
education. Accordingly, any analysis undertaken in relation to these considerations would be
based on assumptions as opposed to concrete evidence. The test instrument at third level is
basedonthePISA(OECD2006) mathematical literacy domain and this assessment utilises
real-world/everyday problems, unlike the contexts typically encountered in school mathemat-
ics. Thus, it is concerned with appropriate decision making and application of mathematical
knowledge. The authors concede that this decision making/application may have taken place
through the medium of English, through the medium of Gaeilge or a combination of both,
even though the test was administered through the medium of English. Further analysis by
the authors looks at this aspect but it is too long to include in this paper also, and it is
deserving of a more detailed discussion, which this paper would not facilitate. The
bilingual aspect that the authors are concerned with in this paper is in establishing a
statistical relationship between each language (English and Gaeilge) and the performance
on mathematical word problems through the medium of English/Gaeilge for each
group of students, which has been done in the first half of Section 5.4.Thenext
bilingual aspect to be looked at is to establish a statistical relationship (if it exists) between
language proficiency in both languages and performance on mathematical word problems
through the medium of English. This is reported in the second half of this section.
Table 4 Correlations between mathematics performance (in English) and Gaeilge language proficiency
Pearson’s correlation Significance (p) Description
Primary–second level BG–T1: r=0.651 <0.01 Moderate correlation
BGt–T1: r=0.706 <0.01 Strong correlation
BGc–T1: r=0.605 <0.01 Moderate correlation
All correlations are highly significant.
Second–third level BG–T2: r=0.226 >0.05 Weak to moderate correlations but they
are not significant.BGt–T2: r=0.470 >0.05
BGc–T2: r=0.462 >0.05
Mathematics through the medium of Irish 53

The final analysis of the data looks at the different language proficiency groups—high/
high, low/low and dominant (either in Gaeilge or in English). The dominant group also
includes the monolingual English students. Cummins’Thresholds Hypothesis (1976) does
not distinguish between languages, but argues for the effect, either positive or negative in
cognitive outcomes, of the interplay of languages. None of the bilingual students were
found to be dominant in English at the primary- to second-level transition. Thus, those
categorised as dominant at this transition were dominant in Gaeilge or were monolingual in
English. At third level, some of the bilingual students were dominant in English and others
were dominant in Gaeilge, so within the dominant category, three different types of students
are present.
From Fig. 1, it is obvious that Gaeilgeoirí with relatively high proficiency in both
languages performed better mathematically than students dominant in one language
(Gaeilge and monolingual students), and better than those with low proficiency in both
languages. Mann–Whitney Utests in each case showed that the difference in mathematics
performance on the word problem tests is significant between the High/High proficiency
group and Low/Low proficiency group, between the High/High and Dominant proficiency
groups and between the Dominant and Low/Low proficiency groups (see Table 5).
What is also worth highlighting here is that Gaeilgeoirí dominant in Gaeilge performed
slightly better than the monolingual students, which is consistent with the correlations
found between performance on mathematical word problems and language proficiency in
Gaeilge (see Table 6). However, this difference was not found to be statistically significant.
Language Proficiency Group
DominantLow/LowHigh/High
Percentage received on Maths Word Problems
100.00
80.00
60.00
40.00
20.00
0.00
17
Fig. 1 Comparison of language
proficiency groups and perfor-
mance on mathematical word
problems (in English) at second-
level education
Table 5 Significance of the differences between the means of the language proficiency groups on the
mathematics word problem test (in English) at second-level education
Comparison groups Non-parametric test Significance
between groups
Outcome
High/High vs. Low/Low Mann–Whitney U 0.001 p<0.05, therefore there is a significant
difference between the means.
Low/Low vs. Dominant Mann–Whitney U 0.041 p< 0.05, therefore there is a significant
difference between the means.
High/High vs. Dominant Mann–Whitney U 0.002 p<0.05, therefore there is a significant
difference between the means.
54 M. Ní Ríordáin, J. O’Donoghue

Given that this dominant group outperformed their monolingual peers, it merits further
investigation into the mathematics register through Gaeilge and whether this register and
the Irish language facilitates Gaeilgeoirí’s understanding of mathematics word problems at
this transition stage. The most at-risk group consists of the Gaeilgeoirí with low proficiency
in both languages, as all other groups significantly outperformed this group mathematically.
Similar findings were revealed at the transition to third-level education (Fig. 2). Once
again, bilingual students with a high level of proficiency in both languages outperformed
their monolingual peers, and those dominant in one language. Equally, the bilingual
students with low proficiency in both languages performed poorly on the mathematical
word problems in comparison to all other groups.
Significant differences in performance on mathematical word problems were found
between the High/High and Low/Low proficiency groups, and between the Dominant and
Low/Low groups (Mann–Whitney U). All other differences were not statistically significant
(see Table 7).
However, it is worth noting that the bilingual students who were dominant in English
performed slightly better than their monolingual peers and better than bilingual students
dominant in Gaeilge (Table 8). This suggests that these students had not reached the
threshold level necessary in Gaeilge in order to reap the cognitive benefits from being
bilingual evident for those with high proficiency in both languages The monolingual group
in turn performed better than the bilingual students dominant in Gaeilge. Therefore, the
greater level of English-language proficiency may facilitate a stronger transfer of
mathematical skills to the new language of learning (English) at third level for Gaeilgeoirí.
Table 6 Summary of the average performance of the language proficiency groups on the mathematics word
problem test (in English) at second-level education
Language proficiency group Mean of mathematics word problem test
High/High 72.62
Low/Low 51.25
Dominant in Gaeilge 67.03
Monolingual 60.27
Language Proficiency Group
DominantLow/LowHigh/High
Percentage received on Maths Word Problems
80.00
60.00
40.00
20.00
0.00
Fig. 2 Comparison of language
proficiency groups and perfor-
mance on mathematical word
problems (in English) at third-
level education
Mathematics through the medium of Irish 55

5.5 Discussion of findings
Overall, these findings demonstrate that Gaeilgeoirí’s performance on mathematical word
problems is related to their linguistic proficiencies in both languages. For primary-level
Gaeilgeoirí in the transition to English-medium second-level mathematics education,
Gaeilge language proficiency (the language of learning) was found to be of more
significance than proficiency in English. Also at this transition, Gaeilgeoirí’s performance
on the English version of the mathematics test was highly correlated with their performance
on the Gaeilge version of the test. This is consistent with Cummins’Developmental
Interdependence Hypothesis (1979a), which proposes that the greater the level of academic
language proficiency in a student’s first language, the stronger the transfer of skills across to
the new language of instruction. This suggests that Gaeilgeoirí with a high level of
proficiency in Gaeilge performed well due to a strong transfer of mathematical skills across
to English. At this transition in Irish education, when assessed through the medium of
English, Gaeilgeoirí, in this study, experienced a disadvantage of 8.7% in performance on
mathematical word problems. Improving language proficiency in English may improve
Gaeilgeoirí’s performance in mathematics through the medium of English. However, the
challenge lies predominantly with the mathematics teacher when assessing these students. It
may be beneficial to undertake assessment in a student’s first language of learning until
adaptation to the new language of instruction has taken place to ensure that assessment is
valid. For Gaeilgeoirí in the transition to English-medium third-level education, a more
significant relationship was found between English-language proficiency and performance
on mathematical word problems. This is perhaps due to the more decontextualised nature of
the mathematics word problems utilised and their reliance on independent decision making
and the application of appropriate mathematical knowledge. Gaeilgeoirí with low
proficiency in English (and in Gaeilge), on average, performed the poorest on the
mathematical word problems.
Table 7 Significance of the differences between the means of the language proficiency groups on the
mathematics word problem test (in English) at third level education
Comparison groups Non-parametric test Significance
between groups
Outcome
High/High vs. Low/Low Mann–Whitney U 0.041 p<0.05, therefore there is a significant
difference between the means.
Low/Low vs. Dominant Mann–Whitney U 0.039 p< 0.05, therefore there is a significant
difference between the means.
High/High vs. Dominant Mann–Whitney U 0.38 p>0.05, therefore no significant
difference between the means.
Table 8 Summary of the average performance of the language proficiency groups on the mathematics word
problem test (in English) at third level education
Language proficiency group Mean of mathematics word problem test
High/High 72.04
Low/Low 35.35
Dominant in Gaeilge 46.21
Dominant in English 62.62
Monolingual 57.07
56 M. Ní Ríordáin, J. O’Donoghue

Clearly, differences exist between the two transitions in Irish education. The Gaeilge
language is of more significance at the primary- (Gaeilge medium) to second-level (English
medium) interface, whereas English-language proficiency has a stronger influence on the
transition from second- (Gaeilge medium) to third-level (English medium) education. The
findings suggest that developing mathematical literacy through the medium of Gaeilge at
primary level will enhance the transfer to English-medium mathematics education at second
level. However, this is not followed through at Gaeilge-medium second-level mathematics
education. The findings at this transition provide support for developing mathematical
literacy through the medium of English at second-level education in order to facilitate the
transition to English-medium third-level mathematics education. The differences in the two
sets of data imply that a change in language emphasis occurs over time and that learning
through the medium of Gaeilge at primary level and through the medium of English at
second level may enhance mathematical learning for Gaeilgeoirí. However, it is important
to keep in mind that the number of participants at each transition is relatively small,
particularly at the second- to third-level transition and, thus, the findings need to be inferred
with caution.
However, the most significant overall finding at both transitions is the support for the
Thresholds Hypothesis by Cummins (1976). In both transitions, language proficiency
groups were identified and those with a high proficiency in both languages outperformed
their monolingual peers, those dominant in one language and those with low proficiency in
both languages. Also, bilingual students displaying low proficiency in both languages were
mathematically weak and lagged behind their peers. These results are consistent with the
findings of Dawe (1983) and Clarkson (1992) who also draw on the work of Cummins.
Clearly, Gaeilgeoirí face the challenge of developing an adequate proficiency both in the
English and Gaeilge languages, as high proficiency in both may enhance mathematical
performance on word problems as suggested by the findings.
This is the first Irish investigation into the area of mathematics education and
bilingualism, and the first national study undertaken in the area of mathematics and the
Gaeilge language. Although MacNamara (1966) carried out a study previously in Ireland,
his work was concerned with students from an English background forced to learn through
the medium of Gaeilge. This research project is different in that it is concerned with
Gaeilgeoirí in the transition from Gaeilge-medium to English-medium mathematics
education. Also, significant methodological flaws were found in the work carried out by
MacNamara (see Cummins 1977). Thus, the originality of this research lies in the fact that
it is the first research of its type carried out in the Irish educational context. The key
findings presented in this paper provide support for the Thresholds Hypothesis by Cummins
(1976). The authors’work replicates that of Dawe (1983) and Clarkson (1992), whose
findings provide evidence and support for the Thresholds Hypothesis by Cummins (1976)
within a mathematics education context. Thus, the authors’work is contributing to the
robustness of international findings, while validating the importance of Cummins’work in
relation to bilingualism and mathematics education. This hypothesis has been investigated
extensively at primary-level education, a little at second-level education but rarely at third-
level education. Thus, the work undertaken by the authors contributes to the development
and support of this hypothesis beyond primary level given that the research was undertaken
at second and third-level education in Ireland. Very little research has been undertaken in
the area of mathematics learning and bilingualism at third-level education (Neville-Barton
& Barton 2005). Therefore, this research provides a contribution to this area of research in
mathematics education, as well as providing a foundation for future research to be carried
out.
Mathematics through the medium of Irish 57

The findings of this study demonstrate that language proficiency and mathematics
education are related. The work carried out reveals that bilingualism is not a problem and
may enhance mathematics performance on word problems if Gaeilgeoirí have developed an
adequate proficiency in both languages. First-language proficiency (Gaeilge) is a key aspect
for success in mathematics learned in a second language (English) at second-level education
in Ireland for Gaeilgeoirí. This reinforces similar findings from other countries such as
Scotland (Johnstone, Harlen, MacNeil, Stradling & Thorpe 1999) and Wales (Williams 2002)
where both Heritage Maintenance Language and Immersion Education are established.
However, at second-level Gaeilge-medium education, a language shift occurs, and for
Gaeilgeoirí transferring to English-medium third-level education, English-language profi-
ciency is of more importance for a successful transition. This is consistent with findings from
similar educational contexts such as New Zealand (Neville-Barton & Barton 2005)where
second-language learners experienced a disadvantage of between 10–15% in mathematics
learning due to English-language difficulties (their second language). A characteristic feature
of both transitions in the Irish context was that low proficiency in both languages could be a
significant factor in hindering learning in mathematics for Gaeilgeoirí, but given that this
research was undertaken at transition points, further investigation is needed to assess if these
students adapt to their new learning context and catch up at a later stage.
5.6 Implications for teaching and learning
Although the sample of students involved in this study is relatively small from which to draw
generalisable conclusions about all Gaeilgeoirí in the transition to English-medium
mathematics education, we consider that the findings reported here present a good description
of language proficiency and its influence on performance on mathematical word problems for
bilingual students in Ireland. These students are in a particular situation—switching from
learning mathematics through the medium of Gaeilge to the medium of English. The data was
collected at a particular stage in time, for two different sets of students. There are slight
differences in the two sets of data which may imply some changes occur over time. A number
of implications and applications for the teaching and learning of mathematics for bilingual
students can be suggested at each transition stage. These include the following:
&Performance on mathematical word problems is related to language proficiency. Therefore
on entering second or third-level education, bilingual students’proficiency in English and
in Gaeilge should be assessed. This may be useful for teachers in order to identify students
of high proficiency, as well as those who have low proficiency, so as to cater for their
learning needs at the initial transition to a new language medium for learning mathematics.
&For Gaeilgeoirí in the transition from primary- to second-level education a significant
relationship exists between their performance on mathematical word problems in
English and their proficiency in Gaeilge. By adopting a functional view of language,
teachers can enable students to develop mathematical literacy (understand and talk
about mathematics using the mathematics register) through the medium of Gaeilge and
employ this literacy to understand practical experiences (Mohan & Slater 2005). By
developing a sufficient level of mathematical literacy through the medium of Gaeilge it
should assist the transition to English-medium second-level mathematics education for
Gaeilgeoirí (Cummins’Developmental Interdependence Hypothesis 1979a).
&For Gaeilgeoirí entering English-medium third-level education there is a high
correlation between a student’s performance on mathematical word problems in
English and their proficiency in English. Therefore, it may be appropriate for second-
58 M. Ní Ríordáin, J. O’Donoghue

level mathematics teachers to assess the language proficiencies of Gaeilgeoirí in upper-
second-level Gaeilge-medium education in order to identify Gaeilgeoirí with low
proficiency in English. By providing partial instruction through the medium of English
and developing bilingual mathematics learning resources, it may develop Gaeilgeoirí’s
mathematical literacy in English and, thus, assist these students’transition to English-
medium third-level mathematics education.
&Collaboration between mathematics departments and language departments should be
fostered so as to provide for the optimum development of mathematical literacy in
unison with language proficiency.
Addressing the needs of Gaeilgeoirí in the transition to English-medium mathematics
education is of paramount importance to this research project. In order to cater for
Gaeilgeoirí experiencing this transition, appropriate teaching interventions are indicated so
as to enhance the mathematics competence and the language proficiency of these students,
and to allow the potential cognitive benefits of being bilingual to come to the fore. Further
research is needed both to confirm these results with larger groups and in other locations,
and also to investigate more fully issues of causality.
Acknowledgements This research is funded by the Mathematics Applications Consortium for Science and
Industry (MACSI), through Science Foundation Ireland (SFI), and in conjuction with the National Centre for
Excellence in Mathematics and Science - Teaching and Learning (NCE-MSTL) at the University of
Limerick. Any opinions, findings, conclusions or recommendations are those of the authors and do not
necessarily reflect the views of the foundations. The authors are grateful for the comments from various
reviewers during different stages of writing the paper.
Appendix A—A Selection of Mathematics Word Problems (Second-Level Education)
A Selection of English Questions:
Question 3
5 is a factor of 20.
The factors of 20 are {1, 2, 4, 5, 10, 20}
The factors of 28 are {1, 2, 4, 7, 14, 28}
In these two sets of factors, a number of factors are common to both sets—1, 2, 4. The
highest of these, called the Highest Common Factor (or H.C.F. for short) is 4.
Write out the factors of 10 and 15 and hence find the H.C.F. of 10 and 15.
Factors of 10: ______________________
Factors of 15: ______________________
H.C.F. _______
Question 5
12 is a multiple of 3 as we can multiply 3 by 4 to get 12. 12 is also a multiple of 4, of 6 and
of 12. 18 is a multiple of 2, of 3, of 6, of 9 and of 18.
List the next 6 multiples of 3 which are greater than 2.
Question 11
In a class of 30 girls, 17 play tennis and 15 play netball. If all the girls play at least one of
these games, how many girls play both?
Mathematics through the medium of Irish 59

A Selection of Gaeilge Questions:
Ceist 2:
Scríobh síos na huimhreacha seo leanas ag baint úsáid as uimhireacha:
(i) Dhá chéad agus caoga ocht __________________________
(ii) Trí mhíle, ceithre chéad agus seachtó ocht __________________________
(iii) Ochtó sé __________________________
(iv) Deich míle, ceithre chéad agus cúig __________________________
Ceist 8
Le linn treimhse de trí uaire thit an teocht i mBaile Átha Cliath ó 6°c go-3°c, agus i Nua
Eabhrac thit an teocht ó 18°c go 10°c. Cé acu des na cathracha a bhraith an ladhdú is mó
san teocht?
Baile Átha Cliath
Nua Eabhrac
Ceist 10
Ag úsáid luibiní {}, liostaigh na heilimintí des na tacair seo leanas:
m.s. An tacar dos na laethanta don tseachtain a thosnaíonn le D {Deardaoin, Domhnaigh}
A= An tacar dos na slánuimhreacha réidh idir 11 agus 25. _________________________
B= An tacar dos na gutaí san teanga Béarla. ___________________________
C= An tacar dos na iolraí de 5 idir 8 agus 37. ___________________________
D= An tacar dos na séasúir don bhliain ___________________________
E= An tacar dos na dathanna i soilse trachta. ___________________________
Appendix B—A Selection of Mathematics Word Problems (Third-Level Education)
Question 2
For a college assignment you monitored your daily activities for a week. The activities and
amount of time spent doing them are represented in the table below.
Type of activity Time
Watching TV Over 3 h/day
Studying 4 h
Sport 0.5 h
Socialising 2–6 h/day
Computer 1 h a few days a week
Would you represent this data on a bar graph? Give a reason for your answer.
Ans.
60 M. Ní Ríordáin, J. O’Donoghue

Question 5
Susan wishes to build a fence around a rectangular lawn. The lawn is 50 m long and 30 m
wide. One long side of the fence will be made of stone and the other three sides will be
made of wood. Stone costs €6 a metre, and wood costs €3 a metre. How much will the
fence cost Susan?
Ans. ________________
Question 7
Sarah was preparing to go to America for the duration of the summer holidays. She needed
to change some Euros (€) into American Dollars ($). The exchange rate between the Euro
and the American dollar was:
1 Euro ¼1:21 American dollars
1. Sarah changed 1500 Euro into American dollars at this exchange rate. How much
money in American dollars did Sarah get?
Ans. ________________
On returning to Ireland after 3 months, Sarah had $700 left. She changed this back to Euros,
noting that the exchange rate had changed to:
1 Euro ¼1:26 American dollars
2. How much money in Euros did Sarah get?
Ans. _______________
3. During these 3 months the exchange rate had changed from 1.21 to 1.26 American
dollars per Euro. Was it in Sarah’s favour that the exchange rate now was 1.26
American dollars instead of 1.21 American dollars, when she changed her American
dollars back to Euros? Give an explanation to support your answer.
Ans. __________________________________________________________________
Question 9
If the length of a square is increased by 10%, and the width of the same square is decreased
by 10%, then the area of the square
A: decreases by 10%
B: decreases by 1%
C: is unchanged
D: increases by 10%
E: increases by 20%
Mathematics through the medium of Irish 61

Question 13
A lighthouse sends out light flashes with a regular fixed pattern. In the diagram below is the
pattern of a certain lighthouse. The light flashes alternate with dark periods.
It is a regular pattern. After some time the pattern repeats itself. The time taken by one complete
cycle of a pattern, before it starts to repeat, is called the period. When you find the period of a
pattern, it is easy to extend the diagram for the next second or minutes or even hours. In the
diagram below, makea graph of a possible pattern of light flashes of a lighthouse that sends out
light flashes for 30 s/min. The period of this pattern must be equal to 6 s.
Please fill in the missing words in the following questions. Only one word is required in
each of the spaces provided.
Question 17
The numbers 0, 1, 2, 3,..... are called whole numbers or _______________. So 75 is an
_______________ but 4 1/3 is not an _______________.
Question 18
Any whole number is divisible by itself and 1. If p is a whole number greater than 1, which
has only p and 1 as factors, then p is called a _______________ number. 2, 3, 5, 7, 11, 13,
17, 19 and 23 are all _______________. 14 is not a _______________ since it is divisible
by 2 and by 7.
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