The Lexis of Maths Lectures
The Lexis of Maths Lectures: The Creation of a Pedagogic Corpus and Wordlist from a Series of Maths Lectures
Alister Drury
Rachel Perkins
Warren Sheard
Language Centre, School of Languages, Cultures and Societies, University of Leeds
ABSTRACT
Researchers have investigated discipline-specific and academic vocabulary in multiple academic disciplines through the creation of word lists (Dang, 2018; Gilmore and Millar, 2018; Watson Todd, 2017; Valipouri and Nassajii, 2013). However, there are currently no mathematics wordlists based on spoken corpora that are suitable for the context in question. The present study involved the creation of a context-specific corpus to investigate the frequency of technical and sub-technical vocabulary in a series of mathematics lectures through the creation of a keyword list. The keyword list was created using Sketch Engine from a 152,443-word corpus of 46 mathematics lectures. The final wordlist comprised 202 lemmas, covering 12.89% of the corpus. The benefit of creating a context-specific wordlist was clear. The New Academic Word List (NAWL) (Browne et al., 2013a) provided just 4.51% coverage of the corpus. Assuming students have knowledge of the first 2000 word families of the New General Service List (NGSL) (Browne et al., 2013b), which provided 84% coverage, total coverage was nearly 97% with the mathematics wordlist, compared to 88.67% with the general academic list. Of the 202 lemmas in the mathematics wordlist, 116 were sub-technical, meaning that they are polysemous, with both a general meaning and a mathematical meaning. This is a feature of the language of mathematics and may be an added challenge for students.
KEYWORDS: corpora, wordlists, technical vocabulary, maths, EMI, polysemous, sub-technical
INTRODUCTION
A growing number of students whose first language is not English are choosing to study on higher education courses in their own country which are taught in part or exclusively in English. These degree programmes are often provided by universities from English speaking countries that have established branch campuses overseas. The number of these transnational education providers (TNEs) has grown rapidly in recent years, with China being a major centre for this expansion (Fang and Wang, 2014).
Students studying on undergraduate degree programmes in these contexts often report difficulties in understanding their subject lectures due to a lack of discipline specific terminology (Soruç and Griffiths, 2018, p.42). In recent years researchers within the field of English for Specific Purposes have sought to address this challenge by producing word lists of discipline-specific corpora in order to better understand variations in lexis between different disciplines and to aid the development of English language teaching materials and curriculum design. Bondi (2010, p.3) defines keywords as ‘those whose frequency (or infrequency) in a text or corpus is statistically significant, when compared to the standards set by a reference corpus’. While lists have been produced for a number of disciplines such as engineering, medicine and agriculture (Coxhead, 2018), the influence these lists have on curriculum design remains unclear (Nation, 2016, p.172). One reason for this may be that these keyword lists are generally derived from relatively large corpora which aim to represent a whole field of disciplinary study and hence do not adequately address the specific needs of curriculum designers. For example, a list of engineering vocabulary may not be of equal utility to electrical engineering as civil engineering.
The problem of disciplinary lexis can be exacerbated in the case of foundation year programmes in which English language instruction is often separated from content instruction delivered through an English Medium Instruction (EMI) model, meaning English teachers are often left unaware of the language challenges facing their learners in their content modules (Galloway and Ruegg, 2020, pp.5-6). Opportunities to provide language support to these learners who struggle to adapt to the high vocabulary load within the content-based courses can, therefore, be missed.
This study aims to bridge this gap through the creation of a key word list derived from a small corpus (152,443 tokens) of mathematics lectures in order to better understand the nature of the lexical challenge facing the students and to aid in the development of pedagogical materials and interventions in order to aid students’ comprehension of these lectures. The corpus and keyword list were based on a series of recorded lectures delivered by an English-speaking lecturer to Chinese students studying at a TNE provider in China.
BACKGROUND
The context of the study
SWJTU-Leeds Joint School is a TNE partnership between the University of Leeds and South-West Jiaotong University based in Chengdu, China. Students study on undergraduate degree programmes in four subjects within the Faculty of Engineering and Physical Sciences (civil engineering with transport, mechanical engineering, computer science and electronic and electrical engineering). At the end of their studies students receive an undergraduate degree from both the University of Leeds and South-West Jiaotong University. Courses last for a period of four years with the first year being common to all strands before students specialise in year 2. During the first year, students study an English for Engineering module taught jointly by English teachers from the University of Leeds and SWJTU as well as content-based modules in physics and mathematics. The content-based modules are taught using English as a medium of instruction (EMI).
The present study focuses on the lectures delivered by a mathematics lecturer from the University of Leeds whose first language is English. Before 2020, this module was taught in person; however, the pandemic has resulted in the replacement of live lectures and classes with recorded lectures through which the content is exclusively delivered. Two of the stated course objectives/learning outcomes of the module are for students to both understand the language of Mathematics and to be able to use this language.
RATIONALE
Lexical Challenge
EMI has seen a rapid expansion in China and indeed throughout the globe in recent years (Dearden, 2015, p.2) and the number of students in transnational education (TNE) is now 1.4 times the number of international students studying in the UK (Universities UK, 2020, p.20). The rationale for EMI/TNE is that through studying a subject in English, students will make gains in both their content knowledge and their language proficiency. However, little research has been done to validate this claim in a HE environment (Macaro et al, 2018, p.57) and the question remains as to what extent studying in a second language impedes content knowledge acquisition (Chang, 2010; Shohamy, 2010, in Doiz et al., 2012). Studies investigating student perceptions of EMI courses have suggested that lack of vocabulary knowledge may be a significant impediment to the comprehension of subject course materials and lectures (Evans and Green, 2007; Chang, 2010; Tatzl, 2011, cited in Harada and Uchihara, 2018).
Moreover, research suggests that adequate comprehension of spoken discourse depends on students knowing a high percentage of the vocabulary in a text. Stær (2009) found that 94% coverage was required to gain a 60% comprehension of advanced level listening texts while 98% coverage was needed for students to obtain a 70% score in a comprehension test. Similarly, van Zeeland and Schmitt (2013) found that adequate comprehension of spoken discourse required students to know between 95% and 98% of the words in a text. Coverage also appears to vary depending on the text type. For example, lower levels of coverage have been found for audio-visual material (Durbahn et al., 2020), dialogues (Giordano, 2021) and texts graded for intermediate level language learners (Noreillie et al., 2018). Another consideration in assessing the difficulty of academic spoken discourse is the number of words learners require to achieve a certain degree of coverage. Dang et al. (2014) found significant variation between disciplines with medical and life sciences requiring 5,000 words to achieve 95% coverage, compared with 3,000 words for social sciences. The present study used screen-capture video lectures. While the visual dimension of the lectures may help to reduce the coverage required for adequate comprehension, the content of the lectures was clearly technical, and they were delivered in the form of monologues. Hence, coverage of between 95% to 98% for the mathematics lectures was assumed would provide adequate comprehension.
Defining General, Academic and Technical Vocabulary
There are different categories of vocabulary in the corpus used in the present study. General, high-frequency vocabulary in English is commonly defined as the most frequent 2000 word families in English, although there is an argument for increasing this to 3000 word families (Schmitt and Schmitt, 2014). A word family consists of the headword (e.g., ‘power’), inflected forms (e.g., ‘powered’) and derived forms (e.g., ‘powerless’). Academic vocabulary, defined as words that ‘have wide range and high frequency in academic texts’ (Dang et al., 2017, p.6), such as ‘analyse’, has been seen by some researchers as separate from these high-frequency words; for example, Coxhead’s Academic Word List (2000), which omitted words included in the General Service List (West, 1953). However, this does not take into account those high frequency words that also have a specialised meaning in a discipline, such as ‘interest’. Therefore, some general academic wordlists; for example, the Academic Vocabulary List (Gardner and Davies, 2014) include this high-frequency vocabulary.
General academic vocabulary is distinct from technical vocabulary. As Ha and Hyland (2017) note, there is no single precise definition of technical language, although there is general agreement on a very specialised use within a discipline and infrequency outside the discipline. There are different categories of technical vocabulary (Dang, 2020, p.439); fully technical words that are known and used by specialists in the field (e.g., ‘arctangent’ in mathematics), lay-technical words that are understood by non-experts (e.g., ‘multiply’), and polysemous words with both a general meaning and a specialised meaning in the discipline, like ‘factor’, whose mathematical meaning is very different from the general meaning. Technical vocabulary is often cited as a barrier to understanding lectures and coursebooks (Evans and Morrison, 2011, p.154; Evans and Green, 2007, p.13). Indeed, being able to understand technical vocabulary is key to understanding of the discipline (Woodward-Kron, 2008, p.246)
As mentioned above, vocabulary items that are commonly found on general service lists of the top 2000 or 3000 most common words in English, or indeed general academic word lists, may also have very specialised meanings in particular disciplines. This type of polysemous lexis, which is referred to as sub-technical (Mudraya, 2006) or crypto-technical (Fraser, 2009), may pose significant difficulties for learners; such technical words may be disregarded by learners if the general meaning is already known (Fraser, 2009, p.157). Accessing the technical meaning of polysemous words may also be problematic (Watson Todd, 2017).
Therefore, in order to fully understand the lexical challenge facing learners, the degree of technicality of vocabulary items in a corpus needs to be analysed (Ha and Hyland, 2017, p.36) in addition to consideration of the number of lexical items or word families that students need to know to achieve sufficient coverage of a text or corpus. An example of a study that has done this would be Fraser (2009), who identified and produced counts for what he termed crypto-technical vocabulary across the various sections of IMRaD-style (introduction, methods, results and discussion) medical research articles.
Identifying technical vocabulary
A number of methods have been employed to identify technical vocabulary in a corpus (Ha and Hyland, 2017). One way is to use the corpus-comparison approach. This method involves comparing a specialised corpus with a general English corpus. Words appearing only in the specialised corpus are assumed to be technical and words that meet a certain threshold of comparative frequency are likely to be technical (Coxhead, 2018, p.8). However, some technical words may not be included using this approach, as polysemous words that have a technical meaning but also have a general high-frequency meaning may not be identified as relatively frequent in a discipline (Ha and Hyland, 2017, p.37). Collocations with a technical meaning may also be omitted (Kwary, 2011). Another method for identifying technical vocabulary is to conduct a keyword analysis. This method also involves the comparison of two corpora. However, in the case of keyword analysis a statistical measure is used to compare the frequencies of words in order to determine words that occur with an unusually high frequency (Coxhead, 2018, p.9). This method generates a keyness score which allows vocabulary items to be ranked. A further method involves making reference to specialised vocabulary knowledge. This may involve consulting specialists in the field or referring to specialist dictionaries in order to distinguish between technical and general or sub-technical vocabulary (Coxhead, 2018; Ha and Hyland, 2017). However, the criteria for inclusion of a word may not be transparent (Nation et al., 2016, p.147), and is subjective (Ha and Hyland, 2017).
The current study used a combination of keyword analysis and consulting specialist vocabulary knowledge. A key word list was first generated to broadly identify likely technical vocabulary before the list was categorised into different subtypes (technical, sub-technical and lay technical vocabulary). It was felt this method was efficient and was sufficiently objective.
Technical and sub-technical language in the discourse of mathematics
While polysemous language is a feature of academic texts in general (Coxhead 2016, p.181), it is particularly relevant to the discourse of mathematics. O’Halloran (2015) characterises this discourse as multimodal, integrating the language of mathematics, its symbolic notation (including superscript and subscript notation), and its graphs and diagrams. While symbols and diagrams may be universally understood, the complexity and technicality of the language of mathematics may present a greater challenge. Grammatical aspects of mathematical discourse include the way that logical relationships are expressed, using conjunctions in a precise way and the use of complex noun phrases, resulting in lexical density (Wilkinson, 2019, p.88). Regarding lexis, Halliday (1975, cited in Wilkinson, 2019, p.88) described the mathematical register as including phrases, such as ‘complete the square’, compound words, such as ‘output’, and words of Greek and Latin origin such as ‘parabola’. Most significantly for the current study, there are also ‘everyday words interpreted in the context of mathematics’ (Wilkinson, 2019, p.88).
It is important to provide support regarding technical and sub-technical language. Students who are not proficient speakers and who may not have a good understanding of technical vocabulary in the context of mathematics may face difficulties on their university courses (Bedore, Pena, and Boerger, 2011, cited in Wilkinson, 2019, p.88). Text comprehension (in both listening and reading) is linked to knowledge of technical language, with students in some disciplines facing considerable challenge; Chung and Nation (2003) identified 37.6% of an anatomy text as being technical, and 16.3% of an applied linguistics text. There is also a link between knowledge of technical vocabulary and content knowledge. Bond (2020, p.104) notes that acquisition of disciplinary vocabulary is ‘a key aspect of gaining access to target knowledge’. As well as having receptive knowledge of this vocabulary, students also need to be able to use it to become part of their discourse community (Wray, 2002, cited in Coxhead, 2016, p.178; Szudarski, 2018, p.140).
Specialised key word lists
The present study aims to help students with the high vocabulary load of their mathematics lectures through the creation of a key word list. The main purpose of wordlists is to identify key vocabulary and provide a target for vocabulary learning, allowing students to see their progress, which can be very motivating (Coxhead 2016, p.180). This is true of all wordlists, including interdisciplinary academic wordlists, such as the Academic Word List (Coxhead, 2000), the Academic Vocabulary List (Gardner and Davies, 2014), and the Academic Spoken Word List (Dang et al., 2017). These interdisciplinary wordlists assume a common core of academic vocabulary across disciplines and can be useful in EGAP contexts (Dang et al., 2017, p.2; Coxhead 2018, p.22). In contrast, a more specialised wordlist focusing on a particular discipline can be even more beneficial for specific contexts. Such a wordlist may be seen as clearly relevant to the student’s course and result in increased motivation (Hyland, 2016, p.20). It provides greater coverage of discipline-specific lexis (Fraser, 2009; Ward, 2009; Dang, 2018) and may result in more efficient learning of lexis (Hyland 2016, p.20).
A wordlist can be specialised in different ways: discipline- or genre-specific, or based on either a spoken or written corpus, for example. Using Becher’s classification of academic disciplines into hard and soft (Becher, 1989), there are several wordlists available for hard sciences, such as Lei and Liu’s (2016) New Medical Academic Wordlist, the Pharmacology Word List (Fraser, 2009), the Basic Engineering Wordlist (Ward, 2009), and the Engineering English Word List (Hsu, 2014). The closest list to one for mathematics is a list of academic lexical bundles (Alasmary, 2019) but this consists of phrases such as if and only if or a set of all rather than individual items of lexis.
While the above are derived from written corpora, there are far fewer wordlists based on spoken corpora. Examples include the Academic Spoken Wordlist (Dang et al., 2017), and the Academic Formulas List (Ellis and Simpson-Vlach, 2010), neither of which are discipline-specific. At the time of the current study, there is one wordlist based on a spoken corpus that is specific to hard science disciplines. Dang (2018) created the Hard Science Spoken Wordlist, based on a corpus of 6.5 million words from six hard-pure disciplines and six hard-applied disciplines, from a range of lectures, seminars, labs and tutorials. Regarding discipline-specific spoken wordlists, there is the Medical Spoken Word List (Dang, 2020), but there is very little that is specific to mathematics derived from either spoken or written corpora.
Bond (2020) discusses the need for specificity in EAP language support, working with disciplines in the university, studying the discourse of these communities and increasing awareness of the importance of discipline-specific language. It was felt that a general wordlist would not meet the needs of our specific context. Therefore, the primary aim of the study was to create a ‘pedagogic corpus’ (Willis, 1990, cited in Szudarski, 2018, p.108), comprising all the language that students encounter on a module. Such a context-specific corpus is the most effective way to target key lexis and reduce vocabulary load for the students (Hyland and Tse, 2007, p.251). A study by Hou (2014) showed that a pedagogic corpus in a different context was used in the creation of learning materials which were successful in improving students’ understanding of disciplinary vocabulary. From our pedagogic corpus, a keyword list was created to form the basis of materials designed to help SWJTU-Leeds Joint School students with the linguistic demands of their mathematics lectures.
METHODOLOGY
Creation of the corpus
Due to Covid, all the lectures used in the creation of the corpus we pre-recorded and thus mp4s were readily available. Permission to use these was obtained from the lecturer involved. Transcripts of the video lectures were created using Otter.ai (a commercially available online transcription application). These were then manually checked against the recordings for lexical and orthographic errors. Punctuation was only corrected in cases where errors impeded comprehension.
The 46 lectures were divided almost equally between the three researchers and each lecture was listened to while reading the transcript. Once corrected, the transcripts were searched using a text editor, Notepad ++, to correct typographic errors and to eliminate inconsistencies with hyphenation (pre-factor, three-dimensional), UK/US spellings (meter/metre, recognise/recognize, labeled/labelled), and compounds of words (arcsecant/arc secant, workout/ work out).
Finally, Sketch Engine was chosen to analyse the corpus data and produce a keyword lemma list, conforming to Bauer and Nation’s level 2 of word families (1993). A lemma consists of a headword and inflections that are the same part of speech. For example, the lemma ‘factor’ as a noun would include ‘factors’ (plural). ‘Factor’ as a verb would include ‘factors’, ‘factored’ and ‘factoring’ in the lemma. Sketch Engine, however, combines headwords that are the same form but a different part of speech, so ‘factor’ as a verb and a noun are the same lemma.
Creation of the keyword list
Webb (2021) suggests that choice of unit of counting for wordlists (word type, lemma, word family) should be made based on the purpose of the list. As such, the lemma was chosen as the unit of counting when analysing and creating our keyword list. This is because research suggests L2 learners have sufficient knowledge to cope with lemmas but often lack morphological awareness to deal with word families (Brown et al., 2022, p.600; Gardner and Davies, 2014, p.30). Lemma was also helpful as several words in the list only occurred in a derived form, such as variable(vary), recursion(recur) and decomposition(decompose). The list would also be needed for both productive and receptive uses, in which case, lemmas are the preferred unit (Nation, 2016, p.26).
The simple maths formula (Kilgarriff et al., 2014) used in the keyword analysis feature of Sketch Engine allows for a variable to be set to change the focus of the list between more common and rarer items (Sketch Engine, no date). Higher values will bias the results in favour of frequency. In our corpus a list created with the variable set at one gives ‘sine’, which has a frequency in the focus corpus of 607, as the word with the highest keyness score. However, if the variable is set to 1000, ‘minus’, whose frequency is 1381, appears first in the keyword list and ‘sine’ appears eighth. The default score in Sketch Engine of one was chosen for our analysis.
As the focus of this research was on technical language rather than features of spoken English, a corpus containing only samples of spoken discourse was selected as the reference corpus, using the spoken component of the British National Corpus (BNC) (2014). Experiments using the full BNC corpus revealed a bias towards idiomatic language and colloquialisms, whereas using the spoken corpus removed these items.
The first version of the list contained a significant number of items of mathematical notation (x, y and dx for example) as well as proper nouns (Chengdu, Leeds) and numbers (one, two, three). As these either did not represent English words or would already be familiar to the students, they were added to a non-word list and excluded from the analysis. A keyword list containing only real words was then obtained.
Refining and categorising the keyword list
Once the keyword list was generated in Sketch Engine it was categorised into four groups: technical vocabulary, sub-technical words with both technical and non-technical meanings, lay-technical words, and a final group comprising high-frequency words with no technical meaning (see appendix 2). This was initially done individually by the three researchers and then any words where there was disagreement were checked together. To be categorised as technical, words had to be monosemous and included in the mathematical dictionary. These included terms such as cosecant, cotangent, and calculus. Words were categorised as sub-technical if they were assigned a general meaning in the Oxford Learners Dictionary (Oxford, 2021) and also had a technical meaning in the Oxford Concise Dictionary of Mathematics (Clapham and Nicholson, 2014). Examples of these terms include derivation, dummy, and log. Finally, words were categorised as lay-technical if they had a mathematical meaning but would be easily understood by a general audience. These words appeared in both the general and mathematical dictionaries and the mathematical meaning was the most common one. Examples of lay-technical items include multiply, radius, and decimal. Words that did not fit into the technical, sub-technical or lay-technical groups and were to be found on the top 2000 most frequent words in the Browne NGSL were then excluded from the list. The decision to exclude general words was justified on the basis that most students would be familiar with these vocabulary items, or they would have been required to learn them in order to pass the College English Test, which requires a knowledge of 4,200 words (Wei, 2004). It should be noted that much research (for example, Lu and Dang, 2022; Sun and Dang, 2020) suggests students at this level in China have not mastered the first 2000 most frequent words. However, as this study was aimed at helping them with their mathematics vocabulary knowledge, it was decided any generally frequent vocabulary would be better dealt with separately in their English lessons. Browne’s New General Service list (2013b) was preferred over other lists as it was readily available in lemmas and in a format which could be used with Range and Ant Word Profiler without any further manipulation. Finally, vocabulary items that appeared five times or less in the corpus were excluded from the list as it was felt that learning these would not be beneficial for students.
The process of categorising the words on the wordlist was not straightforward. Some words found in both the technical and non-technical dictionaries appeared to make reference to the same concept with the main difference being the degree of precision of the definition. An example of this would be variable which is defined as ‘a situation, number or quantity that can vary or be varied’ in the Oxford Learner’s Dictionary but in the Oxford Dictionary of Mathematics as:
an expression, usually denoted by a letter, that is defined for values within a given set. Can be used to represent elements of sets which are not numbers but frequently it relates to numerical quantities and functions defined in them together with the relationship between them.
This problem was further compounded by the fact that some words did not appear in the same form in the mathematical dictionary as in the corpus. For example, the corpus has examples of primed, while the mathematical dictionary only has prime. Another complicating factor was that words often occurred as part of collocations in the mathematical dictionary making direct comparison with the general dictionary difficult.
Once a definitive list of categorised key words had been obtained, it was decided to sub-divide the list. The reason for this was to spread the load of learning for the students. It was felt that words that occurred in all three lecture topics (series, derivatives and integrals) would be best presented to students at the beginning of the course. In order to accomplish this task Range (Heatley et al., 2002) was used to provide data on the number of occurrences in each sub-section of the corpus. The remaining words on the list were then assigned to one of three lists corresponding to each lecture series (see appendix 3).
RESULTS AND DISCUSSION
Categories of vocabulary in the keyword list
The final keyword list comprised 202 lemmas. This were categorised into four categories: technical terms, sub-technical, lay-technical and non-technical. In total 116 words were identified as sub-technical, 35 were found to be technical terms and 20 were designated as lay-technical. The other 31 words on the list were non-technical.
Analysis of the corpus
The mathematics lecture corpus contains 152,443 word tokens (a single occurrence of a word in a text), comprising 2,125 word types (1,496 lemmas).
Table 1 below shows the percentage of words the learners would potentially be able to understand if they knew the first 2000 words of the NGSL (Browne et al., 2013b) and the NAWL (Browne et al., 2013a). The first 1000 would cover just over 79% of the corpus and combined with the second 1000 that would rise to just over 84%. By learning the NAWL they could increase their understanding to nearly 89% of the words in the lectures. Again, the Browne NAWL list was selected because it was readily available in a format which could easily be imported into Range.
These results are broadly in line with other research which shows that a combination of knowledge of the NGSL and the NAWL provides very good coverage of academic texts. However, this does not take into account sub-technical words that possess both a general and a technical meaning; i.e., words where it is unlikely they would know the mathematical meaning despite knowing the more general meaning.
TOKEN TOKEN% CUMTOKEN%
1st 1000 120642 79.12 79.12
2nd 1000 7680 5.04 84.16
NAWL 6879 4.51 88.67
Table 1: Browne NGSL + NAWL
Interestingly, the sub-technical vocabulary on our wordlist accounts for 8.82% of the tokens in the corpus (see Table 2), significantly higher than the coverage provided by the NAWL which suggests that lack of knowledge of this vocabulary may cause learners significant comprehension problems. However, it must be acknowledged that the sub-technical vocabulary was not always used in a technical sense in the corpus. For example, the word ‘even’ was most commonly found in the corpus in its general meaning, while ‘product’ was found exclusively in its technical sense.
TOKENS/% TYPES
Sub-technical 13450/ 8.82 116
Technical 2882/ 1.89 35
Lay-technical 2552/ 1.67 20
Nontechnical 766/ 0.50 31
Table 2: Coverage by word type
Coverage of keyword list
The maths wordlist (as it stands) covers nearly 13% of the vocabulary they encounter in the lectures (see table 3). This means that by learning the first 2000 word families from one of the NGSL lists plus our list will give them approximately 97% of the words needed to understand the lectures. This is a much better coverage than using the NAWL and the NGSL combined.
WORD LIST TOKENS/% TYPES/% LEMMAS
Maths list 19650/12.89 359/16.93 202
Table 3: Coverage of whole list
Sub-technical vocabulary in the corpus
The sub-technical vocabulary in the corpus comprises vocabulary on both the NGSL and the New NAWL, while a number of words are not present in either of these two lists. In total, 56 terms were found to be present in the NGSL, while 33 were in the NAWL. 29 terms were not present on either list (see table 4).
Examples:
Sub-technical vocabulary also on the NAWL: derivative, integral, substitution
Sub-technical vocabulary on the NGSL: function, square, power, value
Neither list: alternate, diverge, inverse
29 were not present in the NAWL or NGSL
56 were in NGSL
33 were in NAWL
Table 4: Breakdown of sub-technical words by NGSL and NAWL
During the process of checking the lecture transcriptions on Otter.ai, it became clear that there was a high number of these ‘everyday words interpreted in the context of mathematics’ (Wilkinson, 2019, p.88). This sub-technical language comprised 8.82% of the tokens in the corpus, compared to just 1.89% for fully technical language. It is very difficult for people with limited understanding of mathematics to judge how similar a word is in its mathematical sense to its general meaning, but some general categories could be discerned.
- High-frequency words (in the 2000 most common word families in English) whose mathematical meanings may be inferred from their general meanings outside the context of mathematics; examples include ‘limit’ and ‘boundary’. (Word frequency was checked using the VocabProfiler function of the Compleat Lexical Tutor site (Cobb, no date), which uses the BNC-COCA frequency lists.)
- Lower frequency words whose mathematical meanings are connected to their general meanings outside the context of mathematics. An example is ‘cusp’.
- ‘Opaque’ (Watson Todd, 2017) or ‘cryptotechnical’ (Fraser, 2009) words which are high frequency outside the discipline. These words have a very different meaning in mathematics, so their meaning cannot be easily understood from their common meanings outside the discipline. (E.g., ‘square’)
- ‘Opaque’ words that are lower frequency, such as ‘differentiate’.
- Words that are used as a different part of speech in mathematics; for example, ‘constant’ and ‘bound’, which are both used as nouns.
- Words with two meanings in mathematics. ‘Prime’, for example, may refer to prime number, or the prime symbol.
- Opaque words that form parts of collocations/multi-word items, like ‘rational function’.
Potential challenges for students and tutors regarding polysemous lexis
There is a lack of literature on the potential language difficulties encountered by Mathematics students, such as those caused by polysemy. In comparing mathematical and general meanings of polysemous lexis for the current study, various potential issues for both students and tutors presented themselves. Students may find searching for mathematical definitions challenging. Often a mathematical definition does not appear in Google Translate or general dictionaries like the Oxford Advanced Learner’s Dictionary or is near the end of the definitions. Specialised mathematics dictionaries such as the Oxford Concise Dictionary of Mathematics include all terms but are not written for English learners. Another issue is collocations and multi-word items. ‘Common denominator’ occurs in Google Translate and is translated into Mandarin, but ‘arbitrary constant’ does not; Google Translate translates each word separately.
Another difficulty is how to approach technical language when the language tutor has no content knowledge. The different categories of sub-technical lexis in mathematics suggest different approaches; knowledge of the non-mathematical meaning can be helpful when the word is frequent and has a connection with the mathematical meaning. Nation discusses core meanings of words and how meanings are often more specific in their technical sense (Nation 2013, p.295 and p.306). Bond (2020, p.101) suggests that such vocabulary may not present significant difficulties for students. However, comparison with the general meaning(s) is less helpful when the word is low frequency and perhaps unknown, or when meanings are very different. If an everyday meaning of a specialised word is known, it may affect comprehension of the same word with a different meaning in a different context (Coxhead, 2018 p.32).
Future directions
One aim of the current study was the indirect application of the corpus in creating teaching materials (as opposed to direct application, when data driven learning is used in the classroom (Flowerdew, 2009; Rohmer, 2011, cited in Szudarski, 2018, p.141)). According to Coxhead (2016, p.117), few studies ‘go beyond simple frequency counts and also consider learnability and teachability’. Teachability is a concern in this mathematics context; using Nation’s (2007) four strands of language learning, tutors working with subject specialists can provide some practice in comprehensible meaning-focused input, using lecture notes and concordance lines to provide context and encourage noticing. Language-focused learning is also possible, with a focus on collocation or pronunciation, for example. A lack of content knowledge on the part of the language tutor means that meaning-focused output and fluency development would require collaboration with subject specialists.
Providing a glossary might be an efficient way of dealing with the issue of technical and sub-technical language in mathematics and might indeed be necessary with lexis that is not easily searchable. However, we hope to use our insights regarding polysemous language in mathematics in the creation of materials to raise awareness of aspects such as opacity and collocation. This awareness would be transferable to other contexts and would hopefully be beneficial to students later in their courses. A further benefit of these materials would be their potential to raise staff awareness of possible language issues for students.
It became apparent during this project that our assumption that the learners would already know the most frequent 2000 words may be incorrect. This has been addressed initially by focusing solely on maths vocabulary, however, we intend to test this assumption during the next academic year as it would have an impact on the content of their English lessons.
Our choice of the Browne NGSL and NAWL is also possibly questionable. It was mainly a choice based on the ease of availability as both lists were available as .xml files in lemma format with inflections included meaning they could be used instantly without needing any further manipulation. As they were used to remove only words of a non-mathematical meaning, we do not believe the choice of list had a meaningful effect on the result.
CONCLUSION
The aim of this project was to build a corpus and use that to analyse the lexical challenges faced by our learners. Clearly, they face a difficult task understanding the lectures from a purely lexical perspective. If we take van Zeeland and Schmitt’s (2013) estimate that learners need to know a minimum of 95% of the words from a lecture to understand it, our learners will not come close to this by learning words from the NGSL plus NAWL (88.67%). We can solve this issue by combining the NGSL with our wordlist, which would allow them to achieve nearly 97% coverage.
The wordlist has allowed the separation of words into those which are technical and therefore outside the scope of EAP tutors and those which are nontechnical or lay-technical that can be taught by EAP tutors. The final category of sub-technical highlights the difficulty of exactly where to draw the line between subject knowledge and language knowledge. Here, the best approach appears to be one of consciousness raising of the learners to the high quantity of potentially opaque language within their discipline and strategies to cope with it.
Addresses for correspondence: a.j.drury@leeds.ac.uk; r.a.perkins@leeds.ac.uk; w.e.sheard@leeds.ac.uk
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APPENDIX 1
Full wordlist by frequency
| 0-51 | 52-102 | 103-153 | 154-202 | ||||
| minus | 1381 | coefficient | 75 | simplify | 32 | manipulation | 13 |
| square | 1263 | pi | 74 | rotate | 31 | intermediate | 13 |
| derivative | 1002 | recursion | 71 | sub | 31 | cotangent | 12 |
| function | 864 | exponential | 71 | increment | 31 | unknown | 11 |
| series | 750 | separately | 70 | extend | 30 | calculus | 11 |
| integral | 739 | true | 70 | finite | 30 | outermost | 11 |
| times | 627 | graph | 69 | quantity | 30 | sufficiently | 11 |
| sine | 607 | irreducible | 68 | subtract | 29 | min | 10 |
| term | 496 | absolute (value) | 68 | equivalent | 28 | max | 10 |
| cosine | 475 | slope | 68 | explicitly | 26 | exact | 10 |
| factor | 453 | chain | 66 | inner | 26 | circumference | 10 |
| power | 367 | arctangent | 66 | individually | 25 | diagram | 10 |
| value | 336 | quadratic | 65 | implicitly | 24 | systematic | 10 |
| evaluate | 317 | equation | 65 | geometric | 23 | stack | 10 |
| give | 298 | convergence | 64 | trigonometric | 22 | bound | 9 |
| dot | 278 | agree | 64 | essentially | 22 | decimal | 9 |
| cube | 263 | polynomial | 62 | proper | 20 | diagonal | 9 |
| constant | 227 | logarithm | 61 | corresponding | 20 | statement | 9 |
| limit | 225 | rational | 58 | geometrical | 20 | branch | 8 |
| converge | 218 | identity | 57 | raise | 19 | parabola | 8 |
| log | 212 | delta | 54 | straightforward | 19 | arcsecant | 8 |
| root | 212 | segment | 53 | verify | 19 | algebra | 8 |
| taylor | 206 | exponent | 53 | obtain | 19 | calculator | 8 |
| formula | 202 | arc | 51 | expansion | 18 | bracket | 8 |
| curve | 186 | contour | 50 | expand | 18 | legitimate | 8 |
| even | 185 | plot | 49 | machinery | 18 | argument | 8 |
| integrate | 183 | alternate | 49 | outer | 18 | transform | 7 |
| substitution | 178 | sequence | 49 | namely | 18 | cusp | 7 |
| expression | 178 | rectangle | 49 | differentiate | 18 | inflection | 7 |
| prime | 170 | strip | 49 | infinitely | 17 | divergence | 7 |
| tangent | 168 | notation | 48 | cosecant | 17 | condition | 7 |
| integration | 166 | degree | 45 | local (max/min) | 17 | accuracy | 7 |
| denominator | 160 | integer | 45 | p-series | 17 | geometrically | 7 |
| infinity | 155 | repeat | 45 | angle | 17 | geometry | 7 |
| multiply | 151 | definite | 44 | valid | 17 | multiplication | 7 |
| secant | 147 | variable | 43 | approximation | 16 | compute | 7 |
| theta | 145 | rid | 42 | continuous | 16 | tricky | 7 |
| fraction | 140 | numerator | 40 | calculation | 16 | specify | 7 |
| product | 135 | derive | 40 | chop | 16 | precise | 7 |
| factorial | 133 | differentiation | 39 | strictly | 16 | improper | 6 |
| partial | 132 | inverse | 38 | maximum | 15 | approximate | 6 |
| convert | 132 | implicit | 38 | radius | 15 | composite | 6 |
| insert | 127 | arcsine | 38 | satisfy | 15 | representation | 6 |
| cancel | 126 | cone | 37 | parameter | 15 | proof | 6 |
| infinite | 120 | endpoint | 36 | conversion | 14 | accurate | 6 |
| interval | 117 | quotient | 35 | common (denominator) | 14 | correspond | 6 |
| ratio | 104 | inequality | 33 | triangle | 14 | non-zero | 6 |
| trig | 96 | substitute | 33 | correctly | 14 | parametric | 6 |
| decomposition | 95 | axis | 33 | preliminary | 14 | manipulate | 6 |
| diverge | 91 | arbitrary | 33 | indefinite | 13 | ||
| linear | 89 | odd | 32 | minimum | 13 | ||
APPENDIX 2
Wordlist by word type and frequency
| Technical | Sub-technical | Lay-technical | General | ||||
| sine | 607 | square | 1263 | minus | 1381 | insert | 127 |
| cosine | 475 | derivative | 1002 | times | 183 | separately | 70 |
| Taylor | 206 | function | 864 | multiply | 151 | true | 70 |
| denominator | 160 | series | 750 | fraction | 140 | strip | 49 |
| secant | 147 | integral | 739 | rectangle | 49 | notation | 48 |
| theta | 145 | term | 496 | cone | 37 | rid | 42 |
| factorial | 133 | factor | 453 | subtract | 29 | inner | 26 |
| trig | 96 | power | 367 | angle | 17 | individually | 25 |
| coefficient | 75 | value | 336 | calculation | 16 | essentially | 22 |
| pi | 74 | evaluate | 317 | radius | 15 | straightforward | 19 |
| recursion | 71 | give | 298 | triangle | 14 | obtain | 19 |
| absolute | 68 | dot | 278 | circumference | 10 | machinery | 18 |
| arctangent | 66 | cube | 263 | diagram | 10 | outer | 18 |
| quadratic | 65 | constant | 227 | decimal | 9 | namely | 18 |
| polynomial | 62 | limit | 225 | diagonal | 9 | valid | 17 |
| logarithm | 61 | converge | 218 | algebra | 8 | chop | 16 |
| arc | 51 | log | 212 | calculator | 8 | strictly | 16 |
| integer | 45 | root | 212 | bracket | 8 | satisfy | 15 |
| numerator | 40 | formula | 202 | multiplication | 7 | correctly | 14 |
| arcsine | 38 | curve | 186 | compute | 7 | preliminary | 14 |
| geometric | 23 | even | 185 | manipulation | 13 | ||
| trigonometric | 22 | integrate | 183 | intermediate | 13 | ||
| geometrical | 20 | substitution | 178 | outermost | 11 | ||
| cosecant | 17 | expression | 178 | sufficiently | 11 | ||
| local (minimum/maximum) | 17 | prime | 170 | systematic | 10 | ||
| p-series | 17 | tangent | 168 | stack | 10 | ||
| Common (denominator) | 14 | integration | 166 | legitimate | 8 | ||
| cotangent | 12 | infinity | 155 | tricky | 7 | ||
| calculus | 11 | product | 135 | specify | 7 | ||
| parabola | 8 | partial | 132 | precise | 7 | ||
| arcsecant | 8 | convert | 132 | manipulate | 6 | ||
| geometrically | 7 | cancel | 126 | ||||
| geometry | 7 | infinite | 120 | ||||
| non-zero | 6 | interval | 117 | ||||
| parametric | 6 | ratio | 104 | ||||
| decomposition | 95 | ||||||
| diverge | 91 | ||||||
| linear | 89 | ||||||
| exponential | 71 | ||||||
| graph | 69 | ||||||
| irreducible | 68 | ||||||
| slope | 68 | ||||||
| chain | 66 | ||||||
| equation | 65 | ||||||
| convergence | 64 | ||||||
| agree | 64 | ||||||
| rational | 58 | ||||||
| identity | 57 | ||||||
| delta | 54 | ||||||
| segment | 53 | ||||||
| exponent | 53 | ||||||
| contour | 50 | ||||||
| plot | 49 | ||||||
| alternate | 49 | ||||||
| sequence | 49 | ||||||
| degree | 45 | ||||||
| repeat | 45 | ||||||
| definite | 44 | ||||||
| variable | 43 | ||||||
| derive | 40 | ||||||
| differentiation | 39 | ||||||
| inverse | 38 | ||||||
| implicit | 38 | ||||||
| endpoint | 36 | ||||||
| quotient | 35 | ||||||
| inequality | 33 | ||||||
| substitute | 33 | ||||||
| axis | 33 | ||||||
| arbitrary | 33 | ||||||
| odd | 32 | ||||||
| simplify | 32 | ||||||
| sub | 31 | ||||||
| rotate | 31 | ||||||
| increment | 31 | ||||||
| extend | 30 | ||||||
| finite | 30 | ||||||
| quantity | 30 | ||||||
| equivalent | 28 | ||||||
| explicitly | 26 | ||||||
| implicitly | 24 | ||||||
| proper | 20 | ||||||
| corresponding | 20 | ||||||
| raise | 19 | ||||||
| verify | 19 | ||||||
| expansion | 18 | ||||||
| expand | 18 | ||||||
| differentiate | 18 | ||||||
| infinitely | 17 | ||||||
| approximation | 16 | ||||||
| continuous | 16 | ||||||
| maximum | 15 | ||||||
| parameter | 15 | ||||||
| conversion | 14 | ||||||
| indefinite | 13 | ||||||
| minimum | 13 | ||||||
| unknown | 11 | ||||||
| min | 10 | ||||||
| max | 10 | ||||||
| exact | 10 | ||||||
| bound | 9 | ||||||
| statement | 9 | ||||||
| branch | 8 | ||||||
| argument | 8 | ||||||
| transform | 7 | ||||||
| cusp | 7 | ||||||
| inflection | 7 | ||||||
| divergence | 7 | ||||||
| condition | 7 | ||||||
| accuracy | 7 | ||||||
| improper | 6 | ||||||
| approximate | 6 | ||||||
| composite | 6 | ||||||
| representation | 6 | ||||||
| proof | 6 | ||||||
| accurate | 6 | ||||||
| correspond | 6 | ||||||
APPENDIX 3
Wordlist by subject and frequency
| Useful to all | Derivatives | Integrals | Series | ||||
| minus | 1381 | integral | 739 | series | 750 | Taylor | 206 |
| square | 1263 | tangent | 168 | converge | 218 | factorial | 133 |
| derivative | 1002 | secant | 147 | integrate | 183 | infinite | 120 |
| function | 864 | theta | 145 | substitution | 178 | diverge | 91 |
| sine | 607 | quadratic | 65 | integration | 166 | convergence | 64 |
| term | 496 | identity | 57 | partial | 132 | alternate | 49 |
| cosine | 475 | delta | 54 | interval | 117 | sequence | 49 |
| factor | 453 | segment | 53 | decomposition | 95 | geometric | 23 |
| power | 367 | arc | 51 | linear | 89 | expand | 18 |
| value | 336 | contour | 50 | recursion | 71 | p-series | 17 |
| evaluate | 317 | plot | 49 | irreducible | 68 | bound | 9 |
| give | 298 | repeat | 45 | absolute | 68 | divergence | 7 |
| dot | 278 | differentiation | 39 | polynomial | 62 | condition | 7 |
| cube | 263 | inverse | 38 | rational | 58 | accurate | 6 |
| constant | 227 | implicit | 38 | strip | 49 | ||
| limit | 225 | arcsine | 38 | rectangle | 49 | ||
| root | 212 | substitute | 33 | degree | 45 | ||
| log | 212 | inner | 26 | cone | 37 | ||
| formula | 202 | implicitly | 24 | endpoint | 36 | ||
| curve | 186 | trigonometric | 22 | inequality | 33 | ||
| even | 185 | geometrical | 20 | odd | 32 | ||
| times | 183 | outer | 18 | sub | 31 | ||
| expression | 178 | angle | 17 | rotate | 31 | ||
| prime | 170 | local | 17 | finite | 30 | ||
| denominator | 160 | chop | 16 | proper | 20 | ||
| infinity | 155 | calculation | 16 | machinery | 18 | ||
| multiply | 151 | radius | 15 | expansion | 18 | ||
| fraction | 140 | maximum | 15 | infinitely | 17 | ||
| product | 135 | preliminary | 14 | cosecant | 17 | ||
| convert | 132 | triangle | 14 | approximation | 16 | ||
| insert | 127 | conversion | 14 | continuous | 16 | ||
| cancel | 126 | minimum | 13 | correctly | 14 | ||
| ratio | 104 | cotangent | 12 | manipulation | 13 | ||
| trig | 96 | outermost | 11 | indefinite | 13 | ||
| coefficient | 75 | unknown | 11 | systematic | 10 | ||
| pi | 74 | calculus | 11 | circumference | 10 | ||
| exponential | 71 | stack | 10 | diagram | 10 | ||
| separately | 70 | diagonal | 9 | min | 10 | ||
| true | 70 | legitimate | 8 | max | 10 | ||
| graph | 69 | branch | 8 | exact | 10 | ||
| slope | 68 | argument | 8 | decimal | 9 | ||
| chain | 66 | arcsecant | 8 | algebra | 8 | ||
| arctangent | 66 | specify | 7 | calculator | 8 | ||
| equation | 65 | cusp | 7 | parabola | 8 | ||
| agree | 64 | inflection | 7 | tricky | 7 | ||
| logarithm | 61 | geometrically | 7 | transform | 7 | ||
| exponent | 53 | geometry | 7 | improper | 6 | ||
| notation | 48 | manipulate | 6 | approximate | 6 | ||
| integer | 45 | composite | 6 | non-zero | 6 | ||
| definite | 44 | representation | 6 | ||||
| variable | 43 | proof | 6 | ||||
| rid | 42 | parametric | 6 | ||||
| derive | 40 | ||||||
| numerator | 40 | ||||||
| quotient | 35 | ||||||
| axis | 33 | ||||||
| arbitrary | 33 | ||||||
| simplify | 32 | ||||||
| increment | 31 | ||||||
| quantity | 30 | ||||||
| extend | 30 | ||||||
| subtract | 29 | ||||||
| equivalent | 28 | ||||||
| explicitly | 26 | ||||||
| individually | 25 | ||||||
| essentially | 22 | ||||||
| corresponding | 20 | ||||||
| obtain | 19 | ||||||
| straightforward | 19 | ||||||
| verify | 19 | ||||||
| raise | 19 | ||||||
| namely | 18 | ||||||
| differentiate | 18 | ||||||
| valid | 17 | ||||||
| strictly | 16 | ||||||
| satisfy | 15 | ||||||
| parameter | 15 | ||||||
| common | 14 | ||||||
| intermediate | 13 | ||||||
| sufficiently | 11 | ||||||
| statement | 9 | ||||||
| bracket | 8 | ||||||
| precise | 7 | ||||||
| multiplication | 7 | ||||||
| compute | 7 | ||||||
| accuracy | 7 | ||||||
| correspond | 6 | ||||||