1English for mathematics (EFM) remains a rather under-researched area in English for specific purposes (ESP). A possible hypothesis to explain the scarcity of studies in this field is the significant cognitive and disciplinary challenge faced by ESP instructors or researchers while dealing with mathematics, which, as opposed to other disciplines such as law, economics or biology, can be highly intimidating and impenetrable for people who have not received advanced instruction in the field. Also, the language of mathematics is highly specific, as it is, somehow, a language in its own right, with sentences including some linguistic elements combined with a high number of mathematical operators, connectors, symbols (O’Halloran 2008). This leads to a very peculiar integration of language and mathematical expressions, as shown by Petit (1991), who found that, while three different constructions may occur with proper nouns in EFM (the ‘s genitive, the ‘of’ genitive and the NØN structure), they were neither always possible nor equivalent, depending on the context of use. In a subsequent article (Petit 1993), he highlighted unusual, specific syntactic patterns associated with abbreviations or the use of proper nouns in EFM.
2Most surveys on EFM have focused on research articles in mathematics (henceforth RAMs). McGrath and Kuteeva (2012) applied Hyland’s stance and engagement theory to a small corpus of 25 mathematics research articles and found a low number of hedges and attitude markers compared to other academic disciplines, while shared knowledge remarks and reader references were more frequent than expected. Their findings also highlighted that writers were well aware of the discoursal norms of their discipline and of the need to adhere to certain conventions. Graves et al. (2014) studied the move-step structure of introductions in 30 RAMs and found significant differences with patterns observed in other disciplines, highlighting that knowledge claims based on rhetorical steps aimed at “establishing presumptions” were a key building block of the genre under study. In another study also inspired by Swales’ genre analysis approach (1990), Moghadassi and Graves (2017) found that introductions of articles in discrete mathematics did include rhetorical steps destined at ‘establishing a niche’ in the landscape of the discipline. Following a similar approach, Onder-Ozdemir and Ozdemir (2019) conducted a contrastive study between a small corpus of 20 RAs in Applied Mathematics and 20 RAs in Economics and determined that the Discussion sections were assigned very dissimilar roles in the two disciplines, requiring highly specific information from specialist informants in the case of Applied Mathematics. Other surveys of EFM focused on specific lexis and phraseology. In an ESP Master’s dissertation, Bouyé (2014) demonstrated the highly formulaic nature of phraseology in research articles dealing with algebra, a finding confirmed by Steidlová (2022).
3As most RAMs are not IMRAD-type papers and tend to be written as mathematical demonstrations based on a limited number of formulae (e.g., “it follows that” or “we can infer that”), it should come as no surprise that EFM includes a significant number of “phrase frames,” i.e., semi-fixed phraseological patterns associated with recurring rhetorical functions and based on a “core” formula, as demonstrated by Cunningham (2017) in a study of 128 RAMs. Among other studies of interest on EFM, McGrath (2016) reviewed online drafting and revision processes in relation to multi-authored RAMs which were collaboratively written on an open-access research blog. Moghadassi et al. (2019) analyzed possible associations between rhetorical moves and the role of visuals in a small corpus of 30 RAMs, underlining the fact that visuals performed ontological, argumentative, and epistemological roles.
4To date, literature on needs analysis in EFM or, more broadly, on the linguistic aspects of mathematics instruction, appears to be very scarce. Of note, there has been some interest for EFM needs in EMI (English as a medium of instruction) contexts, notably in Indonesia, where national legislation has required mathematics instructors to teach in English since 2003, in a major shift from previous regulations prescribing the use of Bahasa Malaysia (as reported by Heng and Tan 2006). Tan and Lan (2011) showed that the return to English in mathematics instruction encouraged the use of translation and simplification strategies (notably the use of keywords for memorization) by Indonesian upper secondary school instructors, thereby widening the gap between strong and weaker students. Nurpahmi et al. (2020, 91) identified some key grammar components cited by a small number of Indonesian respondents as closely associated with topics in mathematics such as plane geometry, functions or derivatives. Farrugia (2009) reflected on code-switching and specific registers between Maltese and English for mathematics instructional purposes in primary schools. Prochazkova (2013) discussed interactions between the language of instruction and that of mathematics in the context of a CLIL (Content and Language Integrated Learning) approach in the Czech Republic. Bensoussan and Golan (1981) developed a specific, comprehensive curriculum and a textbook combining basic disciplinary instruction with specialized language exercises. Apsari et al. (2020) conducted a needs analysis among 27 students enrolled in a mathematics education program at an Indonesian university. Their study aimed to create an EFM manual and the findings revealed specific needs associated with mathematical terms and speaking skills required to pronounce mathematical symbols. Finally, as regards technology-assisted learning in EFM, Malasari et al. (2021) reported positive perceptions towards blended learning (during the Covid-19 pandemic) in a small sample of EFM students. Malasari and Fortinasari (2023) also suggested that the use of Whatsapp for distant learning could improve motivation and participation in a small cohort of EFM students.
5As can be seen, only a relatively small body of studies has covered some aspects of EFM to date. To the best of our knowledge, no study so far has investigated the needs of students and academics willing to report the results of their research in scientific conferences and peer-reviewed journals, notably in the form of abstracts, talks or research papers in mathematics.
6At École normale supérieure Paris-Saclay (henceforth ENS), students specializing in mathematics receive instruction totalling eighty-five teaching hours overall in both English for general purposes (henceforth EGP) and EFM in the first three semesters of their curriculum. Since 2016, ENS has developed systematic in-house language certification procedures for assessing the scientific writing skills of its undergraduate students. These efforts have included the deployment of SWAP (the Scientific Writing Assessment Program), a multidisciplinary certification focusing on key skills in English for Research and Publication Purposes, henceforth ERPP (see Saber et al., 2020).While SWAP was designed to cater for the needs of the majority of ENS students, students in mathematics required a more specific approach. To meet this need, the ME2 (Mathematical Expression Evaluation), a light standardized test was developed by Professor Saber based on classroom experience and input from informants at ENS’s Department of Mathematics (see Appendix). After two years of initial deployment, English language instructors at ENS felt the need to obtain feedback on the test’s formula, paving the way for a possible revision of test procedures if needed, and to perhaps extend the test to additional EFM skills. Accordingly, the purpose of this study was to further assess the needs of mathematics students and researchers, with a view to developing novel evaluation and certification procedures in EFM.
7Serafini et al. (2015) underlined the need for more rigor and consistency in studies trying to ascertain the needs of English language learners. To increase their reliability and the validity of their findings, recent needs analysis surveys have tried to adhere to the mixed-methods approach set forth by Long (2005). Consistent with this approach, we relied on several sources of information, with a view to “triangulating” the language needs of EFM learners. As suggested by Long (2005, 28), “triangulation is a procedure long used by researchers, e.g., ethnographers, working within a quantitative, or naturalistic, tradition to help validate their data and thereby, eventually, to increase the credibility of their interpretations of those data.” The triangulation of needs in this study relied on data obtained from two sources, i.e., a questionnaire survey and semi-guided interviews conducted between February and April 2023 at ENS Paris-Saclay, France.
8The questionnaire survey was divided into two sections, consisting of twenty-three questions in total. Section 1 sought to obtain information about the sample of participants and comprised questions pertaining to their linguistic profiles and academic backgrounds. Areas of academic specialty, envisaged career paths and personal aspirations were also covered in Section 1. Section 2 assessed the perceived importance of skills related to EGP and EFM in their future careers as doctoral students and academics and comprised sixteen questions. To prepare the questions, a set of target skills and sub-skills associated with English proficiency was defined by the investigators, encompassing both oral and written comprehension and expression, across various professional, mathematical, and academic contexts and settings (see table 1). The set of skills and sub-skills was selected based on input from our extensive teaching experience of EFM at ENS Paris-Saclay and on specific indications and suggestions from our colleagues working as instructors and researchers with the Department of Mathematics at ENS. We aimed to cover a very extensive set of skills to cover most of the professional situations in which students would use English in the future, after graduating from ENS, with an emphasis on research contexts.
Table 1. Skills and sub-skills covered in the Questionnaire Survey
|
Skills
|
Sub-skills
|
|
Needs related to EGP
|
Grammar, Listening, Speaking, Reading, Writing, Interactions/Cultural skills
|
|
Needs related to EFM
|
Non-transparent mathematical terminology
Vocabulary describing mathematical phenomena and graphs
Vocabulary for logical reasoning and linking
Vocabulary related to topical subjects in the field of mathematics
|
|
EFM speaking skills
|
Reading out numbers, basic mathematical symbols, and mathematical operations
Reading out questions
Avoiding pronunciation mistakes
Carrying out mathematical demonstrations before an audience
|
|
EFM reading skills
|
Understanding scientific articles
Understanding mathematics textbooks in English
|
|
EFM writing skills
|
Writing an abstract
Writing a research article in mathematics
Writing a cover letter for a journal
|
|
EFM listening skills
|
Understanding videos and tutorials about mathematics
Understanding conference talks and lectures
Understanding informal conversations about mathematics
|
|
Research internship-related skills
|
Applying for an internship
Job interviews
Presenting one’s laboratory and research interests (while on an internship)
Writing an internship report
Presenting one’s research to colleagues/supervisor s (e.g. in the context of meetings during an internship)
|
|
Conference/seminar skills
|
Listening to a presentation, taking notes and reacting
Giving a short talk at a conference, including body language
Interacting with the audience (e.g., Q&A)
Participating in group discussions
Presenting an advanced demonstration
|
9Most of the questions were multiple choice questions, some of which allowed participants to provide alternate responses by using response categories such as “other” (i.e., when identifying one’s mother tongue and envisioned career path) and “I don’t know” (i.e., when asked to indicate the envisioned career path and field to specialize in, as well as assessing the perceived importance of sub-skills). Some open-ended questions were also asked to obtain insights into any sub-skills that the participants wished to add. Respondents also had the opportunity to leave free comments at the end of the survey. Likert scales were used in Section 2 to allow students to rate how important they considered various skills and sub-skills. Figure 1 shows an example.
Figure 1. Example of the Likert Scale Used in the Questionnaire Survey
10The objective was to obtain responses from at least sixty participants, to ensure sufficient statistical representativeness and to reflect the views of at least two classes of mathematics students at ENS (typically, each class includes twenty-five to thirty students every year). From February to May 2023, the link to the survey, with a brief cover letter, was circulated using Limesurvey among a total of ninety graduate students receiving training at ENS Paris-Saclay at the time of the survey or recent ENS alumni.
11This study was also based on two sets of semi-structured interviews with six academics affiliated with higher education institutions in the region of Paris, in line with triangulation methods used in other needs analysis studies in such fields such as mountaineering (Wozniak 2010), engineering (Spence & Liu 2013), academia (Smith et al. 2022) or agricultural settings (Contreras & Moore 2022). Triangulation in needs analysis has been consistently recommended, under various forms, by a host of studies (notably West 1994; Jasso-Aguilar 1999; Flowerdew 2012; Huhta et al. 2013; Serafini et al. 2015; Widodo 2017). For each interview, a cover letter, as well as a consent form, were prepared, detailing the objectives of the interviews and the way the collected data would be processed. In the first set of interviews, the six respondents were asked to share their perceptions about the importance of EFM in their field and how they would assess the ME2 examination. In a second set of interviews, the six respondents were asked about their academic background, their current positions and the way they used English as researchers in mathematics. They were also invited to comment on flashcards representing the skills or situations that could be associated with their daily practice of mathematics.
12A total of sixty-one participants participated in the questionnaire survey, among whom an overwhelming majority (98.5%) reported French as their native language, with one student indicating a different mother tongue. Approximately 63% of the participants self-assessed their English proficiency as “advanced” (C1 or C2 Council of Europe levels), and close to 30% as “intermediate-advanced” (B2 level). As these reported levels are based on self-perceptions, it should be noted that they may not be reflective of the actual levels of the participants. Nonetheless, one should note the high level of confidence participants appeared to entertain regarding their proficiency in English as a second language. Only 4.8% described their English proficiency as “intermediate” (B1)” or “beginner-intermediate” (A2).
13As shown in table 2, a majority of participants seemed to have clear plans for the future, both academically and professionally. Firstly, most were able to indicate which field they wished to specialize in: 43.5% planned to specialize in theoretical mathematics and 32.3% in applied mathematics, while 24.2% did not provide a response. A vast majority planned to pursue careers in research (52.9%) or teaching (32.3%). Most planned to pursue careers in research (52.9%) or teaching (32.3%), and a small minority (ranging from 1.9% to 4.8%) in data science, finance, business, and civil service. Those who were still unsure about possible career paths were few (2.9%). In addition, there was a similar proportion of those who had plans to work abroad and those who did not (40.3%, respectively). Finally, almost two thirds of respondents had not received previous training in EFM at the time of the study.
- 1 Note: the highest values appear in italics.
Table 2. Questionnaire survey – overall profile of the sample (percent of respondents).1
|
Self-assessed English proficiency
|
|
|
Advanced (C1/C2 European levels)
|
61.9
|
|
Intermediate-advanced (B2)
|
27.4
|
|
Intermediate (B1)
|
4.8
|
|
Beginner-intermediate (A2)
|
4.8
|
|
Previous training in EFM
|
|
|
No
|
61.3
|
|
Yes
|
37.1
|
|
I don’t know
|
1.6
|
|
Projected areas of research
|
|
|
Theoretical mathematics
|
43.5
|
|
Applied mathematics
|
32.3
|
|
I don’t know
|
24.2
|
|
Envisioned career path
|
|
|
Research
|
52.9
|
|
Education
|
32.7
|
|
Data science
|
4.8
|
|
Finance
|
2.9
|
|
I don’t know
|
2.9
|
|
Business
|
1.9
|
|
Civil service
|
1.9
|
|
Plans to work abroad
|
|
|
Yes
|
40.3
|
|
No
|
40.3
|
|
I don’t know
|
19.4
|
14Participants were first asked to indicate whether they perceived EGP or EFM as more important (see table 3). Close to 42% of the participants chose EGP and 29% chose EFM. Both EGP and EFM were considered to be important by 29% of the sample. A possible explanation of this predominance of EGP may lie in the fact that mathematics students receive a limited amount of EGP instruction in their first years at university or in French classes préparatoires (typically 2 hours out of a total of thirty teaching hours per week), while EGP skills are widely perceived as important for any professional career today.
Table 3. Perceived importance of EGP v. EFM (%)
|
English for general purposes (EGP)
|
41.9
|
|
English for mathematics (EFM)
|
29
|
|
Both
|
29
|
15Furthermore, a set of questions allowed to ascertain the perceived needs in EGP. Participants were asked to rate the importance of each EGP-related skill such as grammar, reading, writing, listening, speaking, as well as interactional and cultural skills (e.g., informal conversations; see table 4).
Table 4. Perceived importance of skills related to EGP (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Grammar
|
6.5
|
16.1
|
30.6
|
35.5
|
11.3
|
|
Listening
|
1.6
|
8.1
|
11.3
|
41.9
|
37.1
|
|
Speaking
|
1.6
|
3.2
|
17.7
|
32.3
|
45.2
|
|
Reading
|
1.6
|
9.7
|
17.7
|
32.3
|
38.7
|
|
Writing
|
1.6
|
4.8
|
25.8
|
40.3
|
27.4
|
|
Interactional and cultural skills
|
3.2
|
9.7
|
27.4
|
30.6
|
31.5
|
16Most EGP skills received high scores from a significant majority of respondents. The mean percentage of respondents who described EGP skills as “important” or “very important” was 67.3% (range 46.8% – 79%), with listening rated first and all four core skills above 67%, while interactional skills and grammar received the lowest rates. Responses rating skills as “not important” were consistently below 10%, with grammar receiving, however, a rating of 16.1%, suggesting lesser interest from participants. Therefore, most participants seemed to consider English as an important subject matter.
17Both reading and writing were perceived as important, but less so than the skills related to oral proficiency. However, a significant minority (25.8%) saw writing as “possibly important,” and those who did not attribute any importance to writing were few (4.8%).
18Grammar was the only skill for which those who felt certain of its importance represented a minority; nevertheless, it was positively perceived overall. It was indeed seen as “important” or “very important” by 46.8% of the sample and as “possibly important” by approximately 30%. Compared to other skills, those who did not see grammar as important represented the largest cohort (16.1%), and those who were unable to respond were also more numerous (6.5%). Hence, although many saw it as potentially useful or quite useful, grammar did not seem to be prioritized by most participants in the context of EGP compared to other needs, an interesting finding which seems to reflect the acceptability of English as a lingua franca (i.e., comprising a certain number of non-standard grammatical features) among our respondents to achieve communication purposes.
19Participants were then invited to express the perceived importance of the needs related to EFM lexis and phraseology (see table 5). These categories, along with those that appear below for other EFM skills, were formulated based both on classroom experience with students and on input from informants affiliated with the department of mathematics at ENS Paris-Saclay.
Table 5. Perceived importance of lexis and phraseology skills related to EFM (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Mathematical terminology (non-transparent relative to French terms)
|
3.2
|
9.7
|
19.4
|
27.4
|
40.3
|
|
Vocabulary for describing visual information in mathematics (graphs, etc.)
|
3.2
|
8.1
|
30.6
|
35.5
|
22.6
|
|
Vocabulary for logical reasoning and linking
|
3.2
|
6.5
|
12.9
|
45.2
|
32.3
|
|
Vocabulary related to topical subjects in the field
|
4.8
|
10.9
|
23.4
|
33.9
|
28.2
|
20On average, more than 62% of respondents rated various skills associated with lexis and phraseology as “important” or “very important” (range 58.1% – 77.5%), which definitely marks out vocabulary as a priority in any EFM curriculum. Vocabulary for logical reasoning and linking was deemed important by the largest majority (77.5%). Although the skill associated with describing visual information in mathematics received fewer votes, approximately 90% of participants viewed the skill as important, including those who marked it as possibly important.
21Three participants also left comments to communicate what they perceived was important in EFM. (1) Knowing which tense to use when writing a scientific article on a mathematical subject, (2) stylistic conventions, and (3) mathematical formulations in English, when different from those in French, were indicated. These comments provide examples of challenges associated with stylistic conventions in mathematical and scientific English that learners face. Participants were then asked to express the perceived importance of various EFM speaking skills (see table 6).
Table 6. Perceived importance of EFM speaking skills (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Reading out numbers, symbols and operations
|
3.2
|
11.3
|
33.9
|
27.4
|
24.2
|
|
Reading out equations
|
3.2
|
11.3
|
30.6
|
30.6
|
19.4
|
|
Avoiding pronunciation mistakes
|
1.6
|
25.8
|
32.3
|
24.2
|
16.1
|
|
Presenting mathematical demonstrations before an audience
|
6.5
|
6.5
|
12.9
|
35.5
|
38.7
|
22While the overall perceptions toward all four skills cited above appear positive, the findings demonstrate that certain speaking skills in EFM tend to be less appreciated (“important” and “very important” range 40.3% – 74.2%). For example, only 19.4% of participants perceived “reading out equations” and “avoiding pronunciation mistakes” as “very important,” and 16.1% rated the latter as “not important.” This could be linked to the fact that the language of mathematics, when displayed on slides or handwritten on the backboard during academic presentations, conference talks, and lectures, is self-explicit for field specialists, and that slips generally do not affect comprehension.
23The trend of associating slightly less importance with speaking skills in EFM can be correlated with that related to grammar skills in EGP. The fact that most participants deemed pronunciation and grammar skills as less important than other skills suggests that their priorities – whether in EGP or EFM – may have more to do with getting their messages across and less with native-like fluency.
24In contrast, 74.2% of respondents evaluated “presenting mathematical demonstrations before an audience” as “important,” or “very important.” Therefore, the ability to carry out a demonstration, which is at the heart of mathematics, seems to be prioritized over any other speaking skills in EFM. This specific skill, which can be broken down into various steps requiring specific linguistic and discursive skills (e.g., defining a variable, reaching a conclusion, etc.), thus appears to be seen as a core EFM competence.
25As shown in table 7, reading skills associated with EFM were highly rated by the majority, with relatively little difference in the perceptions of the two skills evaluated in the survey (“important” and “very important” range 74.2% – 79%).
Table 7. Perceived importance of EFM reading skills (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Understanding scientific articles
|
1.6
|
9.7
|
9.7
|
16.1
|
62.9
|
|
Understanding textbooks
|
1.6
|
14.5
|
9.7
|
24.2
|
50
|
26The premium that participants placed on reading skills suggests that textbooks are frequently employed in the training of mathematics students, and that reading research articles is a frequent practice among mathematicians.
27As can be seen in table 8, perceptions of writing skills followed patterns relatively similar to those observed for other skills, with 74.2% and 87.1% of participants rating “writing an abstract” and “writing a research article,” respectively, as “important” or “very important.” On the other hand, “writing a cover letter,” perhaps a more anecdotal skill, received lower scores.
Table 8. Perceived importance of EFM writing skills (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Writing an abstract
|
1.6
|
6.5
|
17.7
|
38.7
|
35.5
|
|
Writing a research article
|
3.2
|
8.1
|
22.6
|
45.2
|
41.9
|
|
Writing a cover letter for a journal
|
4.3
|
12.4
|
24.2
|
30.6
|
28.5
|
28Participants were also asked to rate listening skills in EFM (see table 9). Despite the overall positive perceptions, there were fewer participants who described these skills as “important” or “very important,” when compared with most of the skills (range 48.4% - 72.6%). However, skills related to understanding conference talks and lectures were appreciated by many, deemed as “important” and “very important” by 72.6% of participants.
Table 9. Perceived importance of EFM listening skills (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Understanding videos and tutorials about mathematics
|
1.6
|
22.6
|
27.4
|
25.8
|
22.6
|
|
Understanding conference talks and lectures
|
1.6
|
11.3
|
14.5
|
35.5
|
37.1
|
|
Understanding informal conversations about mathematics
|
1.6
|
11.3
|
27.4
|
21
|
38.7
|
29As for more complex EFM skills related to research and internships, as can be seen in table 10, all skills were positively perceived by more than half of the sample, but seemingly less prioritized than other skills (“important” and “very important” range 48.4% – 66.1%). “Presenting one’s research to colleagues/supervisors” and “writing a research report” were evaluated as “important” or “very important” by a significant majority.
Table 10. Perceived importance of EFM skills related to research activities or internships (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Applying for an internship
|
1.6
|
16.1
|
19.4
|
33.9
|
29
|
|
Job interviews
|
4.8
|
16.1
|
25.8
|
27.4
|
25.8
|
|
Presenting one’s laboratory and research interests
|
4.8
|
16.1
|
30.6
|
37.1
|
11.3
|
|
Writing a research report
|
1.6
|
8.1
|
24.2
|
37.1
|
29
|
|
Presenting one’s research to colleagues/ supervisors
|
3.2
|
4.8
|
25.8
|
25.8
|
40.3
|
30The findings from this section appear to be correlated with the sample profile, as many participants were enrolled in the first year of a master’s program at the time of the study. At this level, students are likely to have encountered situations in which they have had to communicate their research in the form of reports and presentations and therefore have come to perceive the relevant skills as central to the field of mathematics. In contrast, the competences that are seemingly less – at least directly – related to research and academia may have been perceived less favorably, because students at this level are not likely to have looked for a job in relation to their research interests.
31Although our respondents were either students or recent alumni who did not yet have significant exposure to scientific conferences, skills associated with these events appeared to be highly prized, with four of the five sub-skills exceeding a score of 60% as “important” or “very important,” as can be seen in table 11. This appears to reflect the premium given to research in the mathematics curriculum at ENS.
Table 11. Perceived importance of EFM skills related to scientific conferences and seminars (%)
|
I don’t know
|
Not important
|
Possibly important
|
Important
|
Very important
|
|
Listening to a presentation, taking notes
|
0
|
11.3
|
27.4
|
37.1
|
24.2
|
|
Giving a short talk at a conference
|
3.2
|
6.5
|
29
|
37.1
|
24.2
|
|
Interacting with the audience
|
3.2
|
6.5
|
25.8
|
38.7
|
25.8
|
|
Participating in group discussions
|
3.2
|
9.7
|
35.5
|
33.9
|
17.7
|
|
Presenting a sophisticated demonstration
|
1.6
|
12.9
|
24.2
|
37.1
|
24.2
|
32We conducted six semi-structured interviews with researchers in the field of mathematics, affiliated with higher education institutions in Paris. Some of the profile traits of the sample are highlighted in table 12. Interviewees majored in mathematics or followed a double curriculum in mathematics and neuroscience, mathematics and civil engineering, or mathematics and computer science, and most specialized in applied mathematics.
33The use of English was prevalent in the professional lives of the specialists: all respondents described English as crucial when collaborating with international researchers or PhD students, and four respondents used English frequently when reading and writing research articles, and two respondents in professional settings, overall. They emphasized that, in France, most research articles in the field of mathematics are written in English, especially in applied mathematics.
Table 12. Semi-structured interviews – sample profile
|
Branch of mathematics
|
N
|
|
Applied
|
5
|
|
Fundamental
|
1
|
|
Mother tongue
|
|
|
French
|
5
|
|
Italian
|
1
|
|
English language skills (self-reported)
|
|
|
Excellent
|
1
|
|
Very good
|
2
|
|
Good
|
2
|
|
Intermediate
|
1
|
34Respondents perceived skills related to both EGP and EFM as important. However, four stressed the importance of EGP when expressing complex ideas and nuances and communicating with non-specialists. Several respondents also reported that skills related to EFM were “limited,” naturally acquired through professional exposure.
35When asked to identify about the hallmarks of EFM, formal register, terminology and phraseology, set formulas and linking words, were mentioned. Across several cases, the simplicity of syntax and style in EFM were accentuated (e.g., “a very small dataset of words” and the highly standardized and formulaic nature of mathematical research writing).
36Then, respondents were asked to comment on the importance of specific language skills in mathematical settings. Skills related to lexis and phraseology (i.e., non-transparent mathematical terminology, vocabulary related to mathematical phenomena and graphs, vocabulary for logical reasoning and linking, vocabulary related to topical subjects in the field) were highly valued, especially those related to reasoning and linking skills and describing mathematical phenomena. In contrast, respondents expressed that field-specific vocabulary was less important, as it is often acquired naturally through exposure.
37Respondents agreed that all speaking skills (i.e. carrying out mathematical demonstrations, presenting one’s work environment and research interests, presenting one’s work progress, and giving classes) were very important. However, there were diverging opinions about reading out numbers, mathematical symbols, operations and equations: some saw it as a fundamental competence, while some saw it as a secondary objective, given that numbers and symbols often appear on a slide or written document. Listening skills were seen as important in the contexts of conference meetings and informal interactions, but less in understanding videos and tutorials. As for interaction skills (i.e., leading an informal discussion on mathematics, conducting interviews, taking part in a round table discussion, answering questions from the public, etc.), informal discussions and answering questions were identified as two essential and frequently mobilized skills.
38In reading, understanding research articles in mathematics, reports, doctoral dissertations and textbooks was important in the everyday professional lives of the respondents. Textbooks were seen as easier to read due to frequent exposure. Some experts added that reading skills were required for occasional uses of user manuals. Moreover, writing abstracts, articles, emails and applications for research funding was a regular practice for the respondents. The importance of writing research articles, abstracts and emails was stressed.
39It should be noted that responses appeared to be strongly correlated with the level of English of each respondent: skills that were mentioned more frequently tended to be those that were least mastered. Additionally, the responses seemed to be influenced by the working context of the interviewees, as oral and listening skills were rated more important for respondents working in international settings, and reading and writing skills were more valued in everyday tasks and local contexts. What all respondents agreed on is that overall English language skills are important in their careers, especially writing skills.
40In the last section of interviews, we presented the ME2 (Mathematical Expression Evaluation), ENS Paris-Saclay's current in-house program for testing EFM skills, to the respondents, and asked them to comment on the three tasks assessed in the test (foundation skills, editing skills and expression skills, see appendix).
41The first task (Foundation skills: pronunciation of mathematical symbols, equations and expressions) was deemed helpful in conference participation and teaching. However, many respondents expressed that reading out equations extensively was not a usual occurrence, and, even in such cases, they managed to find ways to fill terminological gaps by, for instance, using Latin words or visual aids.
42Thus, several respondents described the first task as less central and complex than tasks 2 and 3 and suggested lowering the weighting associated with the task. Some recommended that we ask students to read and comment on the use and significance of an equation or an algorithm, rather than solely testing their ability to read mathematical notions out loud, attributing more importance to the context around the equation or algorithm. The editing skills and oral demonstration skills targeted in the second and third tasks (Editing skills: improving the linguistic quality and the clarity of a draft abstract and Expression skills: presenting a short mathematical demonstration) were highly valued by all researchers.
43Lastly, respondents identified a competence related to free expression and argumentation skills they believed was not targeted in the test. They were extremely concerned to see young researchers struggle to express and structure ideas coherently. The ability of adapting register and style to professional contexts was also seen as important. Therefore, some respondents suggested adding a writing section in the form of a report, research proposal or email to the test.
44A key finding of this study was that English language skills were, overall, highly rated by French mathematicians. This was not a foregone conclusion, given the fact that the French school of mathematics is one of the dominant powers on the international scene, with 13 Fields Medal recipients since 1950. Further, many prestigious French peer-reviewed journals in mathematics (Annales scientifiques de l'École normale supérieure de Paris, Journal des Mathématiques Pures et Appliquées, Bulletin de la Société mathématique de France, Annales de l’Institut Henri Poincaré etc.) continue to publish articles in French, thereby allowing researchers to partly bypass the predominance of English as the lingua franca of research which is seen in many other disciplines.
45Some language skills were more prized than others by respondents. In EGP, grammar was perceived as less important than other core skills, notably listening. In EFM, reading out equations or avoiding pronunciation errors received lower scores than reading research articles and textbooks, presenting research results before an audience or writing research reports. Reading skills were often cited as important by respondents in our cohort. Of note, all skills directly related to research in mathematics were highly valued as well. These results may have important implications for syllabus design and testing procedures in EFM in higher education institutions.
46Students, alumni and academics alike all attributed high scores to competences related to reasoning and linking skills, along with the description of mathematical phenomena. Carrying out a demonstration in English was also perceived as very important. This emphasis on skills associated with deductive reasoning suggests that all respondents shared the same awareness of the specific rhetorical features of mathematical discourse, described by Schiappa (2021, 49) as “the argumentative and stylistic modes of persuasions found in written proofs and arguments throughout the history of mathematics.”
47Overall, the way in which the six academics we interviewed perceived the use of English for professional contexts were consistent with the views expressed by the respondents in our questionnaire survey. In particular, both cohorts perceived EGP skills as important, especially those related to oral proficiency. Further investigation is needed to determine the reasons for this specific priority, which may reflect the need to communicate in English in mathematics laboratories which increasingly host international researchers from all over the world.
48The results of this study have concrete pedagogical implications, notably in relation to EFM course design. EFM instructors could emphasize certain elements of general English to improve skills identified as very important, such as oral communication, reasoning, and structuring an argument in written form. Regarding skills specifically related to EFM, courses could be designed to put less emphasis on technical terminology and more focus on contextualizing and applying broader EFM skills. Certain phraseological patterns directly related to mathematical reasoning could also be integrated into EFM syllabuses, including common phrases found in proofs to introduce notation (“Let us denote by Z the set...” “Throughout the proof, K denotes…”), formulate assumptions (“Let P satisfy the above hypotheses”), signpost steps (“We first prove...We now proceed by induction”) and introduce deductions (“This gives...Thus we get…”). Additionally, familiarization with various genres, such as mathematical proofs, reports, conference presentations, as well as academic and scientific texts like abstracts and research articles, would be useful for graduate students in mathematics. This genre-based pedagogy could integrate multimodality and focus, for instance, on essential presentation skills for presenting a proof, including the use of visuals, non-verbal communication (e.g., posture and eye contact), body language and handling questions.
49This needs analysis also provided useful feedback and insights which may support future changes or improvements in ME2, our in-house EFM test. The academics that we interviewed stressed the importance of improving students’ writing proficiency by adding a writing section to our test, based on their experience of writing skills in specific professional contexts, in particular writing emails or applications for funding. This study also confirmed that editing mathematical abstracts and carrying out a demonstration at a mathematical conference, two skills already covered by ME2, are highly relevant skills in EFM and were prioritized by both students and experts of the field. In contrast, the ability to articulate mathematical symbols, equations, and expressions fluently was identified as intermittently beneficial in teaching and research settings. However, it was perceived as less central compared to other skills, as mathematicians often rely on visual aids when presenting equations to colleagues, a classroom, or an audience at a scientific event. Therefore, we will consider giving less weight to the section in which these skills are treated and place more emphasis on skills related to demonstrating proofs and writing.
50This needs analysis offers some insights into new skills that could be tested in the ME2 standardized examination. Reading skills (especially competences associated with understanding scientific articles), writing skills (writing a research article or abstract, emails or applications for funding) and listening skills (understanding conference talks, lectures and informal conversations) were all reported to be necessary in a mathematician’s career. None of these skills is currently tested in ME2. All six interviewees stressed that a writing section should be added to the test, thereby providing an opportunity for students to work on written expression, style as well as logical reasoning and textual coherence. Future interviews with experienced EFM teachers, interactions with experts from other universities which administer standardized ESP tests are warranted to move towards a test that would cover a broader repertoire of learners’ needs in EFM.
51Expert opinions and student perceptions were used to improve the in-house test in the 2023-2024 academic year at ENS Paris-Saclay. First, based on the experts’ emphasis on the importance of writing skills – in particular, structuring an argument in a formal tone – a writing section was added. This section consists of a structured essay based on a question related to the field of study (e.g., To what extent is it possible to eliminate algorithmic bias in machine learning, and how should the trade-offs between fairness and accuracy be navigated in real-world applications?). Second, certain mathematical terms, such as symbols and equations, were described by respondents as “naturally acquired” through exposure. As a result, the grading scheme of the test was adjusted to give less weight to the first section (reading out equations), which was also modified from a recording format to a written format. Third, given the experts’ emphasis on the importance of oral communication skills, an oral component was introduced. In this part of the exam, English instructors collaborated with mathematics instructors to evaluate students’ demonstration of proofs, followed by a discussion, assessing both linguistic and scientific aspects of their performances.
52Input from research in mathematics discourse could also help further overhaul ME2 in the future. Possible changes in the test could be related to four distinct features of mathematics discourse. Information patterning in mathematics research articles is a first specific feature. In contrast with the CARS (“Create a Research Space”) pattern usually observed in research article introductions, which relies on a move-step rhetorical structure, as highlighted by John Swales (1990), article introductions in mathematics have a different structure, notably a greater emphasis on “establishing presumptions” (Graves et al. 2014), and highly specific rhetorical steps when establishing a “niche” for the study (Moghaddasi et al. 2017). A second notable finding from previous studies is the importance of highly formulaic phraseology in mathematical articles (Bouyé 2014; Cunningham 2017; Steidlová 2022). A third area that could be covered in ME2 is the frequent use of multimodality in mathematics research articles (Moghaddasi et al. 2019). Finally, typical syntactic or linguistic features such as specific nominal forms (Petit, 1991), abbreviations and the use of proper names as syntactic operators (Petit 1993), a lower number of hedges and attitude markers but a higher number of markers of shared knowledge or addresses to readers when compared to other disciplines (McGrath & Kuteva 2012), might also be considered as testable features in mathematics discourse. Last but not least, improving the ability of English language instructors to understand, teach, test and certify EFM skills also represents a major challenge, as underlined by Prochazkova (2013).
53A strength of this study lies in the fact that our initial hypotheses about the repertoire of linguistic skills and sub-skills that we initially formulated to conduct the survey were confirmed by our findings. A key finding is indeed that almost all the skills in the questionnaire were overwhelmingly perceived as important, thereby showing that we were successful in pinpointing the key skills of EFM.
54However, this study may have some limitations. First, the questionnaire survey was conducted at a single institution (ENS Paris-Saclay) and the academic respondents to the interviews only worked in institutions and universities in Paris or its region. The cohorts were thus not representative of mathematicians in France as a whole. Also, ENS Paris-Saclay is acknowledged as an elite, research-driven French “Grande École.” Therefore, perceptions of EGP and EFM needs assessed in this study may not reflect perceptions in larger cohorts at larger universities. In particular, the results of the questionnaire survey suggest that ENS students highly valued reading and writing skills associated with understanding and writing research articles in mathematics. Most participants were highly aware of the importance of scientific research in the context of their curriculum, as well as in their prospective career options as future Ph.D. students and/or researchers. It is likely that the institutional setting and in-house academic culture of ENS Paris-Saclay, which both place a significant emphasis on research, may have contributed to these perceptions.
55However, as highlighted by Floch (2012), one should note the leading role played by the French Écoles normales supérieures as centers for excellence in mathematics: “The[se] colleges provide an environment perfectly suited to mathematicians. They recruit the brightest pupils and set the bar high. “The test for getting into the ENS is an examination of natural ability to do maths to a certain level, says Olivier Faron, head of the Lyon ENS.” Given this leading role on the national scene, it is not unlikely that perceptions about EGP and EFM among ENS alumni and academics are in fact shared by the French community of mathematicians at large.
56Another limitation lies in the fact that some additional key skills may have been overlooked in our questionnaire survey. As previously underlined by Montano (2014), Aharoni (2014) and Mauger (2023), mathematicians tend to relish beauty in demonstrations, a beauty akin to that of sophisticated poetry, based on an economy of means, but also on displacement and unexpected twists in language and reasoning. Euler’s identity (eiπ+1 = 0), for example, is often cited as an example of pure mathematical beauty and elegance. The figure of mathematics genius and university professor Charlie Eppes in Numb3rs, a TV series based on input and advice from professional mathematicians, embodies this quest for elegance and beauty in mathematical problem-solving. Therefore, one of the specific features of EFM, which was not tested in our study, may lie in aesthetic skills which make mathematicians sensitive to beautiful symmetries, exotic geometrical figures, or specific values such as the golden ratio.
57In conclusion, our results highlight that EGP and EFM skills are highly valued by both students in mathematics and academics with significant experience in the field. While some competences such as listening in EGP, reading research articles, carrying out demonstrations, presenting research results and writing research reports in EFM appear to be prioritized by respondents, this needs analysis also underscores that a significant majority of the skills tested in the study were often cited as important, which calls for an integrated EGP/EFM approach in course design for mathematicians studying at higher education institutions, who should benefit from training in multiple skills. Further studies on larger cohorts of respondents are warranted to confirm and extend the results of this study.
Acknowledgments
The authors would like to acknowledge students enrolled at ENS Paris-Saclay's Department of Mathematics and alumni for kindly accepting to participate in this survey. They also extend our thanks to the six academics who took part in the semi-structured interviews prepared for this survey.
Disclosures
The authors report no conflict of interest. This study did not receive any funding or external support. The Mathematical Expression Evaluation (ME2) is a not-for-profit, in-house testing program solely used at ENS Paris-Saclay, France.
Authorship
The four authors contributed to the design of the study and to the manuscript. Associate Professor Manon Bouyé (MB), Associate Professor Claire Kloppmann-Lambert (CKL) and Research and Teaching Associate Lindsey Paek (LP) initiated and conducted the study. They designed and administered the questionnaire used to investigate language needs. CKL formulated, conducted and analyzed the six semi-structured interviews with professional mathematics instructors at ENS Paris-Saclay. MB focused on methodological aspects and wrote the Methods section. LP analyzed the data and generated the tables displayed in this article. Professor Anthony Saber (AS) wrote the introduction and some passages of the discussion. He conducted research on available references for the study and contributed methodological advice. He edited and finalized the manuscript. AS is the author of the Mathematical Expression Examination (ME2) described in this article.