1. Introduction
In recent years, New Zealand secondary schools and tertiary institutions have enrolled increasing numbers of students for whom English is an additional language (EAL students). There is, therefore, growing interest in the language requirements for successful study and in programmes that will assist these students.
It is a common perception that students from Asian countries, particularly China, enter the New Zealand education system with good backgrounds in mathematics. Anecdotal evidence has suggested that these students take mathematics in New Zealand because they perceive that this subject is less reliant on language skills, and that they have a good background in mathematics in comparison with New Zealand students of the same age.
Another group of EAL students entering the New Zealand education system comes from the Pacific Islands and must adapt to a new culture. Many people have suggested that the language issue is an important factor for these students in their adaptation to New Zealand schools (see Appendix 4).
This project was undertaken to better understand the relationship between English language and mathematics learning for both groups of students. We were interested in exploring the extent of any difficulties in learning mathematics attributable to low proficiency in English language, and also discovering particular language features that might cause problems.
Some literature exists that explores this issue at the elementary level (for example, Clarkson,
1991; Setati & Adler, 2001), but there is little work at the senior secondary or tertiary levels. In New Zealand, many of the EAL students arrive in our education system in the final years of secondary school or directly into tertiary institutions.
The senior researchers in this project had already undertaken some preliminary research into learning mathematics in English at first year undergraduate level (Barton & Neville-Barton, 2003). The Teaching & Learning Research Initiative (TLRI) has provided an opportunity to extend this research, and to involve teachers from a variety of environments. Researcher and teaching practitioner partnerships were established to encourage teachers to develop as critical professionals reflecting on their practice, in particular, with respect to language issues in their classrooms.
2. Aims and Objectives
The research theme introduced above relates directly to the TLRI principles. Principle One concerns strategic value to education in New Zealand. The research responds to the increasing diversity of students in the New Zealand education system, and attempts to further understand how linguistic diversity affects their learning. The project addresses the inequalities of this situation.
Principles Two, Three, and Four relate to the quality of the research. By building on work of experienced researchers in the team, and investigating a variety of learning situations, this study was designed to reach substantive findings which will directly affect future practice.
Principles Five and Six relate to the role of teachers as researchers. This project needed to promote significant development of all its teachers as researchers.
Therefore, the aims of this research project were:
- to examine the impact and nature of language factors in the learning of mathematics for EAL students;
- to produce recommendations for mathematics teachers of EAL students;
- to produce guidelines for the design of language support programmes;
- and to develop a group of teachers with an interest in language and mathematics and the skills to continue researching this issue.
Research Questions
- It is known that there are disadvantages for EAL students learning in classrooms where English is the language of instruction. Elder (1993) and Graham (1987) estimate the variability in academic performance due to English language ability is up to 10 percent for university students, and that it is higher for humanities and social science subjects in comparison with mathematics or science subjects. However, Barton and Neville-Barton (2003) suggest that the disadvantage due to language may be just as high in mathematics as in other subjects. Therefore the first research question is as follows.
What is the extent of the disadvantage in mathematics learning due to low English language proficiency at the senior secondary and university undergraduate levels?
- There is a considerable literature on the linguistic features of mathematical discourse in English (e.g., Clarkson, 1991; Dale & Cuevas, 1987; Halliday, 1975; MacGregor & Moore, 1991). There is also a literature that examines discourse features in different languages (e.g., Galligan, 2001; Setati & Adler, 2001). However, while features have been described, there has been limited research on the difficulties these cause for mathematics learners, particularly at senior levels. Abedi (2001), for example, is one of a limited corpus, and has a focus on elementary mathematics. Naudé (2003) is one of a group that compares EAL students with native speakers of English (L1 students). However, the different language features affecting students from different foreign language backgrounds has not been directly addressed.
What specific language features cause difficulty for particular groups of senior and undergraduate EAL students learning mathematics, and how do these compare with language difficulties experienced by L1 students?
- Another aspect of this study was to investigate whether it is general language ability or specific technical language ability that is most important in learning mathematics in another language. The tests used for EAL students entering English-speaking educational institutions (for example, the Cambridge International English Language Testing System (IELTS)) are measures of general academic proficiency. The literature on specific mathematical discourse has been referred to above. These two linguistic considerations generated a third question.
What is the relative importance of technical language knowledge compared with general language proficiency in the learning of mathematics at senior secondary and undergraduate levels?
- Earlier research at undergraduate level by Barton and Neville-Barton (2003) indicated that students were unaware of the extent of their disadvantage due to low English proficiency. It is possible that this is a result of a belief that mathematics learning is language free. Such unawareness is a severe limitation to overcoming any language disadvantage, so it is important to know whether it is widespread.
What awareness do secondary and undergraduate EAL students have of the difficulties they face due to low English proficiency?
- Mathematics classes in urban New Zealand schools and all tertiary institutions have significant numbers of EAL students. For example, in Barton and Neville-Barton’s (2003) research at first-year undergraduate level at the University of Auckland, over 60 percent of their large random sample were EAL students. Understanding the issues faced by these students is critical for effective teaching and should translate into the provision of support.
What practical steps can be taken by teachers in mathematical learning environments to ameliorate language effects; and what support services can be provided by educational institutions?
3. Research Design
The Research Team
The study involved an initial research design team of five practitioners with research experience, each from a different institution and a research co-ordinator from a sixth institution. The design team was identified by Pip Neville-Barton and Bill Barton as successful, active mathematics teachers who had had some contact with university post-graduate programmes. Each member of the team co-opted other teachers from their institutions who had an interest in the topic and who wanted to develop research skills. Each group undertook an independent study in their institution, although the topics were closely related under the project aims, and some research instruments were shared.
Seven meetings were held during the course of this project. The research design team of six met initially in December 2003 and again in February 2004 to discuss the research issues, to distribute and discuss the literature, and to draft a timeline and a research plan that would take into account the different ethnographic features of each institution. Before the full team meeting in March, one of the original research design team members had to withdraw from the project. However, at this stage, Elaine Vine from Victoria University expressed an interest in joining the project with teachers from Wellington Girls’ College. This was welcomed.
The total team therefore, comprised 16 people as listed below (initial design team is underlined):
| Unitec, New Zealand | Pip Neville-Barton (applied linguist and project co- ordinator) |
| University of Auckland | Bill Barton, Bob Chan, Chris King, Viliami Latu (mathematics lecturers) |
| Macleans College | Mark Phillips, Bruce Dixon, Vaughan Mitchell (mathematics teachers) |
| Auckland Girls’ Grammar School | Jushi Hu, Anne Blundell (mathematics teachers) |
| Tangaroa College | Sosefina Paletaoga, Rasela Lafaele, Havili Ofamo’oni (mathematics teachers) |
| Victoria University | Elaine Vine (applied linguist), working with. . . . |
| Wellington Girls College | Monica Luxford (mathematics teacher), Marianne Devere (ESOL teacher) |
There were five further full team meetings where selected literature was discussed, research instruments were developed, findings were shared and discussed, and writing up of individual projects began.
Ethics
Ethics approval was granted through the Unitec Research Ethics Committee on 14 February 2004. Information sheets and consent forms were sent to the principals of the schools, to the HOD Mathematics at the University of Auckland, and Head of School of Linguistics and Applied Language Studies at Victoria University of Wellington. Information sheets and consent forms were also distributed to the student participants before the data were gathered.
Research Participants
The study focused on senior secondary students and third year undergraduate levels of mathematics learning. Participants were mainly from Mandarin, Tongan, or Samoan language backgrounds although two studies included English native speakers. (See Individual Study summaries below.)
Methodological Approaches
Two methodological approaches were applied in the project—the investigative approach and the scientific approach.
The investigative approach was used to find out more about the situations of different EAL students learning mathematics. It involved looking closely at individuals’ experiences of mathematics learning in English, observing their classroom situations and looking at their work, asking them questions, and talking to them about their successes and difficulties.
The scientific approach was used to test particular hypotheses that had been developed in earlier research or in the literature. In particular the extent of any disadvantage for EAL students compared with L1 students, needed to be verified in different ways. Also, tests were designed to show whether in fact particular pre-identified features of language actually did cause difficulties amongst particular populations. This involved administration to large enough groups so that statistically significant results and generalisations could be made.
The original research plan was adapted to accommodate the particular interests and skills of the researchers and their student population. Therefore, although the aims of the research were the same for all five studies, the research tools were individualised for each institution.
Although all the studies had both scientific and investigative features, three of the studies were primarily investigative. One examined an area of mathematics and mathematical discourse not previously studied (proof and argumentation in third year undergraduate mathematics), a second focused on Paskifika-speaking students, using the personal experience of the Pasifika researchers, while a third made use of the applied linguistics background of the researchers to conduct indepth interviews with Mandarin-speaking students. Two studies were primarily scientific, one using mathematically matched Mandarin-speaking and English L1 groups to discover the specific features of mathematical discourse that resulted in statistically significant differences, and the other using a bilingual research design to look closely at English-Mandarin discourse differences.
Observations, tests, questionnaires, and selected interviews were all used as data collection techniques.
Individual Study Summaries
Below are brief summaries of the studies conducted in each institution. Fuller reports are attached as appendices. Further information can be accessed on request.
Auckland Girls’ Grammar School
This study involved 40 Years 12 and 13 Chinese Mandarin-speaking students. The project was fortunate to have a bilingual native Mandarin-speaking teacher, Jushi Hu, who was able to write parallel tests in Mandarin and English. These tests were administered in two sittings seven weeks apart. At each sitting half the students did the English and half the Mandarin version, swapping over in the second test. The analysis focused on comparing students’ performance on the Mandarin and English versions of the test. Group interviews were conducted to gather further insight into the test responses.
The study indicated that these students experienced, on average, a 15 percent disadvantage in overall performance in the English test compared to their performance in the Mandarin test. The syntax of mathematical discourse appeared to cause more problems than vocabulary. The teachers were surprised by some of the misunderstandings revealed in the interviews. There was also lower overall performance, indicating that this group of students is not as mathematically competent as expected by their teachers. Interviews revealed that some students had not had the higher level background usually associated with students from China.
Wellington Girls’ College
This study involved 13 Years 12 and 13 Chinese Mandarin-speaking students. The test and administration paralleled that at Auckland Girls’ Grammar School, with a shorter time between tests. It included a self-reporting of students’ understanding of mathematical instruction. Twelve of the students were interviewed individually and in depth. The analysis focused on the nature of their language difficulties and the strategies the students thought would help their learning.
This study confirmed the disadvantage for students when doing mathematics in English, with a difference of 12 percent on this smaller sample. The interviews revealed particular misunderstandings, a narrow understanding of some concepts, and many strategies for coping with their lack of comprehension. The students self-reported only a little difficulty on average in understanding mathematics in English, despite the problems revealed in their test performance and interviews.
Macleans College
This study involved testing 135 Year 13 students from a variety of language backgrounds. The test collected demographic information and Year 12 grades, and tested mathematical syntax and vocabulary, contextual problems, and problems with redundant information. The analysis was restricted to a Chinese group (14 students) and an English group (17 students) with parallel mathematical ability based on Year 12 grades.
Only three of 32 items showed a significant difference at the 1 percent level—on all three items the English group outperformed the Chinese group. On a further six items there was a significant difference at the 5 percent level; two of these were done better by the Chinese group. No other items showed significant difference. An examination of the individual items revealed the main problems for Chinese students were: prepositions, word order, and interpreting context.
Tangaroa College
This study involved observations of two Year 12 mathematics classes, the administration of two questionnaires to the 42 Pasifika students in these classes, and interviews with 16 students. Initial observations and researcher experience led to a hypothesis that vocabulary was the most important issue for these students. The first questionnaire tested this feature, mathematical syntax, and mathematical word problems. The second questionnaire tested specific discourse features in word problems. Students received the questionnaire in English with a translation into their first language.
The study indicated that vocabulary on its own, particularly instructional vocabulary, was not as problematic as anticipated. Rather it was the combination of syntax and technical vocabulary that caused difficulties. Word problems involving implication were the hardest for the students to solve. During the interviews it emerged that low general proficiency in both languages could also be a significant factor in learning mathematics.
The University of Auckland
This study involved observation of two third-year university mathematics courses. Twelve Chinese-speaking students from one course were then asked to self-report their understanding of the course and were tested on specific mathematical items. English language proficiency results were also available for these students. Follow-up testing of two large courses was undertaken in the second semester to confirm the results and to enable a comparison with English L1 students to be made.
Significant differences were found in third-year classes compared with first-year ones, in that mathematical understanding was much more deeply embedded in the language of the lecturer and texts. The result of the initial testing showed that the disadvantage experienced by the EAL students due to language is higher than expected, and was severe for those students with lower English proficiency. All students appeared unaware of their difficulties. The follow-up testing confirmed these results and indicated that the L1 students did not have any language problems.
4. Findings and Limitations
All five studies offered quantitative or qualitative evidence that EAL students suffer a disadvantage in mathematics learning due to language difficulties. The extent of this disadvantage was measured as 12 percent and 15 percent in two of the studies, and this corroborates with earlier work by Barton and Neville-Barton at first year university level. The benefit of the complete TLRI project is that the disadvantage was investigated in several different ways: a bilingual test, mathematically matched English and Chinese groups, mathematical syntax and logic tests, and interview data (Research Question 1).
Four of the five studies sought evidence of the EAL students’ perceptions of their own understanding of English mathematical discourse. All evidence indicated that they do not realise the extent of their difficulties. We suggest that raising their awareness of this issue is a prerequisite for improving the situation (Research Question 4).
All studies reported that students in general performed worse than the teachers/lecturers anticipated. There was evidence from interviews that, contrary to assumptions, some students did not have the background required for senior levels of mathematics.
Three of the studies revealed that both general and technical English were factors in the problems experienced by EAL students. The Macleans College study, with students of uniformly higher mathematical proficiency, indicated that general English was a bigger factor. The Auckland Girls’ Grammar School study, with students of varied mathematical proficiency, reported that the mathematical discourse was a bigger factor. The indication is that the type of language causing difficulty is related to the mathematical proficiency of the student (Research Question 3).
What specific language features cause difficulty? (Research Question 2). The features varied across the studies, and appear to depend on the mathematical level as well as the home language and English language proficiency levels. Each study reports particular items—see appendices. Vocabulary on its own is not the big issue that was anticipated. However it was a component of the difficulty experienced with understanding mathematical discourse as a whole.
As suggested by the literature, prepositions and word order were key features causing problems at all levels. So also were logical structures such as implication, conditionals, and negation, both at senior secondary and third year university levels. Mathematics couched in everyday contexts caused the expected problems.
The three studies with Chinese-speaking secondary students all reported anomalies in the test item using the word “gradient”. This question was the only one answered significantly better when presented in English rather than in Mandarin. It was suggested that this was because this concept was not taught in China, but was a feature of the mathematics courses taken in English. Further research needs to be done to investigate the conditions under which concepts taught in one language transfer to another language.
Some interview data, along with the experiences of the teacher/researchers, indicate that students having difficulty with language “switch off” in class, relying on texts or handouts. They tend to focus on procedures and approach mathematical problems in tests by trying to recognise a suitable procedure without trying to understand the context. For example, the word “less” may produce a response of “subtract” when this is inappropriate. Language difficulties also seem to limit students’ mathematical solving techniques; for example, such students have difficulty drawing a diagram and are restricted to symbolic mode.
Limitations
Although the separate studies in this research are consistent in their broad conclusions, for example about self-reporting little difficulty with English in mathematics while actually experiencing significant problems with the syntactical aspects of mathematical discourse, each study individually has too small a sample to draw broad generalisations.
The levels of disadvantage were tested in different ways; for example, by comparing EAL and L1 matched groups, and by comparing Chinese-speaking students’ ability when the tests were presented in English and in Mandarin. Furthermore, the levels of disadvantage evidenced in the various studies are also consistent (between 10–20 percent). However, the tests used have not been comprehensive. That is, they have not covered the full range of English language presentations of mathematical discourse (for example, no measure of understanding teacher talk was attempted), and also, not all aspects of mathematical content were covered (for example, statistics was not a part of any of the studies).
It should also be noted that this study took place in Auckland at a particular time. It is likely that there is a particular type of EAL student in the study. Thus, for example, the Chinese-speaking students who participated cannot be assumed to be representative of all Chinese-speaking EAL students learning mathematics in English.
The studies were conducted by practising teachers who have a little research experience, but who are not full-time researchers. There are acknowledged shortcomings of research methods.
Despite these limitations, the studies are, we repeat, broadly consistent, and they also confirm previous research at a similar level. We are confident, therefore, in making the recommendations below.
Recommendations & Further Research
Researchers in each of the studies developed their own set of recommendations and suggestions for further research. They are not repeated here—see appendices. However, there are some ideas that are present throughout this project, and which are important enough to be made into general statements with strong support.
There is no doubt in the minds of the research team that this has been a productive study in personal terms (see Capacity Building below). The mode of involving practising teachers in research about aspects of their particular situation with the active involvement of more experienced researchers and administrative backup was very successful.
- Teachers undertaking supported research, specific to their classroom and subject, is an effective mode of professional development, and should continue to be a significant part of any development programme.
The several studies in this project provide strong evidence to back up other research that EAL students learning mathematics in English suffer considerable disadvantages that are not recognised, either by the students or by their teachers. The myth that mathematics is language free can no longer be sustained. EAL students need support in this area, as in others.
- Resources need to be allocated to supporting EAL students in this area (from Pasifika, Chinese, and many other language groups).
Specific aspects of this support indicated by this research are as follows.
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- Better understanding of these students’ language and mathematics proficiency at the time they enter New Zealand classrooms—and hence better placement of these students.
- The development of special courses in English mathematical discourse, with particular focus on making links between mathematical discourse in the students’ home language and in English.
- The development of in-service programmes for teachers to increase their awareness, and to give them strategies, to support EAL students in their classroom.
This project has uncovered “an iceberg” in the words of one of the teachers/researchers. The issue of language in mathematics classrooms has been acknowledged in the past, but the extent of the problem and the specific nature of the difficulties have not been properly investigated. The various studies in this project show clearly that the issue is both complex and situation-dependent. We need to know a lot more about the issues before we can deal with them properly.
- Further research in this area is warranted.
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- Further research is needed into the mathematical discourse of Pasifika languages.
- Further research is needed into the relationships between Mandarin and Cantonese mathematical discourse and that in English, and whether students’ difficulties arise from these relationships.
- The effectiveness of courses for students designed to support their learning of mathematics in English needs to be properly evaluated.
- The feasibility and effectiveness of providing opportunities for students to discuss mathematics in their home language as part of the pathway to learning mathematics in English, needs to be investigated.
These recommendations address the principle of reducing inequalities in education in New Zealand. This is a theme of strategic importance in education in New Zealand emerging from the diversity of our student population.
The recommendations also focus squarely on the teacher. It is only through their increased interest in this issue, and their awareness and understanding of the problems of language in mathematics learning that progress will be made. This project was an example of effective collaboration between researchers and practitioners, in which each was required to communicate clearly their experience, and acknowledge the other’s point of view. These aspects reflect Principles 5 and 6 of the TLRI Priorities.
Capacity Building
As a team, we explicitly discussed the benefits of this research on more than one occasion in our meetings. In addition, towards the end of the project, we gathered written feedback using a questionnaire entitled “Impact of Research on Teaching”.
We were left with no doubt that this has been a positive involvement for the teacher/researchers, and compares extremely favourably with other professional development experiences. These comments apply to all the teachers in the project, not just to a majority opinion. Part of the evidence for this is that all teacher/researchers worked far in excess of the “paid” time allocation for this project. Furthermore, the researchers who are not teachers were left with a significantly enhanced appreciation of the realities of EAL students in mathematics classrooms.
There were four distinct areas in which the project made a difference: teachers becoming better practitioners; teachers being stimulated to work together and to find out more about language issues; teachers wanting to undertake further formal study; and teachers becoming better researchers. What follows is a selection of the evidence, much of it in the teacher/researchers’ own words.
Better Practitioners
I now try to speak slowly and pronounce words clearly.
I write meanings on the board. Make those meanings clear, repetition of these key words is vital. Encourage mathematical discourse amongst the students.
I have seen [the students] more problems more clearly and in detail since I have been doing this project. I pay more attention to words or syntax used in maths problems.
It has made me think very carefully about the instructions I give to my students verbally or on the board.
I am more appreciative of the gaps in their mathematical language, and need to find ways to encourage them to ask for help, or tell me when they do not understand a term used.
Interest and Collaboration
I found sharing extremely interesting.
I feel my job description has expanded to a teacher of English as well.
A big plus has been and continues to be the opportunity to work with colleagues across sectors and across institutions.
I always look forward to [a co-researcher] coming to school and…talk about how the students performed.
Interesting and stimulating. We have scratched the surface and need more.
I want more data.
Further Formal Study
It took me to the level where I think that learning more and having to research more into this issue is very beneficial. It even leads me to pursue further studies into this field.
It has inspired me to start a PGDipSci with aspirations to complete an MSc in maths ed.
Yes, [in the future I want to do] more research to complete my Masters.
Five of the team will be active in postgraduate study in 2005. Three of these are undertaking further research related to their work on this project.
Teachers Developing as Researchers
It’s been really good for me in that it has allowed me to work with other more experienced researchers, but still have an important role to fulfil. It’s given me the opportunity to participate in a supportive environment, and the luxury of having other perspectives to discuss and consider. Also, because the research is bigger than just me, it gives the whole endeavour an extra bit of legitimation and importance.
I feel that we’ve found an iceberg, and now something needs to be done.
Involvement in research has been good for changing practice because there is a focus on my subject. Generic professional development material is limited in its usefulness to me.
This process was longer than much professional development, and grappled with something in depth.
It has been good to be able to ask the students questions about what they are doing and how. Doing it as research means that there is a structure and a concerted effort to come up with thoughtful questions/answers.
5. References
Abedi, J. (2001). The language factor in mathematics tests, Applied Measurement in Eduction, 14(3), 219–335.
Barton, B. & Neville-Barton, P. (2003). Language issues in undergraduate mathematics: A report of two studies. New Zealand Journal of Mathematics, Vol 32, Supplementary Issue, 19–28.
Barton, B., Chan, R., King, C., & Neville-Barton, P. (2004). The mathematical discourse of advanced undergraduate mathematics. In I. Putt (Ed.) Proceedings of 27th Mathematics Education Research Group Conference, (pp.79–86), Townsville: James Cook University.
Clarkson, P. (1991). Bilingualism in mathematics learning. Geelong: Deakin University Press.
Dale, T., & Cuevas, G. (1987). Integrating language and mathematics learning. In J. Crandall (Ed.), ESL through content-area instruction (pp. 9–52). Englewood Cliffs, NJ: Prentice Hall Regents.
Elder, C (1993). Language proficiency as a predictor of performance in teacher education. Melbourne Papers in Language Testing, 2(1), 68–89.
Galligan, L. (2001). Possible effects of English-Chinese language differences on the processing of mathematical text: A review. Mathematics Education Research Journal, 13(2), 12–132.
Graham, J. (1987). English language proficiency and the prediction of academic success. TESOL Quarterly, 21(3), 505–521.
Halliday, M. (1975). Some aspects of sociolinguistics. In E. Jacobsen (Ed.), Interactions between language and mathematical education: UNESCO Report No. ED-74/CONF-808, 64–73, Paris: UNESCO. Reprinted 1978 as Sociolinguistic Aspects of Mathematical Education, in M. Halliday, Language as social semiotic, (pp. 194–204). London: Edward Arnold.
MacGregor, M., & Moore, R. (Eds.) (1991). Teaching mathematics in the multicultural classroom: A resource for teachers and teacher educators. Melbourne: Institute of Education, The University of Melbourne.
Naudé, A. (2003). The influence of second language mathematics teaching on calculus performance. Remarkable Delta: 03 Communications 199–204. Auckland: International Delta Steering Committee.
Setati, M., & Adler, J. (2001). Between languages and discourses: Language practices in primary multilingual mathematics classrooms in South Africa. Educational Studies in Mathematics, 43(3), 243–269.
The appendices for this online version of the report have been removed. However, you can access them here
