Academic lexical bundles in graduate-level math texts: A corpus-based expert-approved list

1 Mathematics, language and lexical bundle research

The role of language in the construction, dissemination and interpretation of mathematical knowledge is undeniably great, as is pointed out by both mathematics educators and applied linguists. Mathematics educators have explored a variety of topics at the intersection between language and mathematics, including the use of formal versus informal vocabulary instruction (Leung, 2005), mathematical concepts and the way they are defined in different discourse settings (Morgan, 2005), the numerous linguistic problems facing non-English-speaking mathematicians (Chan, 2015; Schleppegrell, 2007), the situated characteristics of mathematics as a discipline (Brilliant-Mills, 1994) and the multi-semiotic nature of mathematical register (O’Halloran, 1998, 2005, 2015). Within applied linguistics and English for specific/academic purposes, interest in the study of mathematics has focused on a limited range of topics, including the investigation of the stance and engagement patterns in pure mathematics research articles (McGrath & Kuteeva, 2012), the rhetorical structure of the introduction sections in journal articles published in the domain of mathematics (Graves, Moghaddasi, & Hashim, 2014), the organization and content of research articles in pure mathematics (Kuteeva & McGrath, 2015), and the processes underlying an online collaboration of authors to draft, revise and publish research articles in pure mathematics (McGrath, 2016).

Two key studies have particularly focused on lexical bundles in mathematics. The study by Herbel-Eisenmann, Wagner and Cortes (2010) was conducted to identify stance bundles occurring in a corpus of classroom interactions. The analysis of stance patterns showed that these patterns served to signal desire (e.g. if you want to), obligation (e.g. you don’t have to) or intention (e.g. you are going to). Another exploration of mathematical discourse was carried out by Cunningham (2017) who extracted frames with fillable slots from a corpus of journal articles and subjected them to structural and functional classification. The structural forms of the frames were then grouped into noun, preposition and verb categories, following similar structural patterns identified in several previous studies. Functionally, the frames were categorized into two key groups: aboutness (e.g. the proof of *) and coherence (e.g. in this section *).

2 Bundle listings

The interest in the study of lexical bundles has encouraged researchers to create several bundle lists, most of which are built with a recommendation for pedagogical use (Ackermann & Chen, 2013; Durrant, 2009; Hsu, 2014; Liu, 2012; Martinez & Schmitt, 2012; Shin & Nation, 2008; Simpson-Vlach & Ellis, 2010; Wood & Appel, 2014). One of the earliest attempts to suggest a list of recurrent clusters was made by Shin and Nation (2008) who focused on patterns occurring in the spoken section of the British National Corpus (BNC). The scope of the study was wide ranging, covering bundles of different lengths and types: two words (e.g. you know) three words (e.g. at the moment) four words (e.g. thank you so much) and frames with fillable slots (e.g. it seems (N, A, to INF, that SV). The study by Wood and Appel (2014) was intended to explore the most recurrent lexical bundles in 1.6 million-word corpus of first-year engineering and business textbooks. The final elements on the list were then searched in a 15,839-word corpus of textbooks for English for academic purposes (EAP) to determine whether common EAP textbooks paid attention to these bundles. A major finding was that a great number of bundles obtained from the disciplinary corpus did not appear in the comparison corpus, an outcome that indicated poor representation of these key clusters in the academic texts. A further examination of the bundles occurring in the specialized corpus revealed that none of them ‘is being presented as teachable units, or highlighted for the reader in any significant way’ (Wood and Appel, 2014, p. 8). On a larger scale, Hsu (2014) applied a set of criteria such as semantic opaqueness and non-compositionality to derive recurrent bundles from a 20-million word corpus of college textbooks, representing 40 different academic domains. A list of 475 sequences was retrieved and then subjected to structural and functional analysis. Bundles took different structural forms, including phrasal verbs (e.g. account for, cope with), passive fragments (e.g. be accustomed to), and prepositional phrases (e.g. in the light of). Functionally, the academic constructions were found to serve as referentials (72%), discourse organizers (22%), and stance expressions (6%).

Another key list was compiled by Ackermann and Chen (2013) who used a combination of corpus methods and expert intervention to build a list of collocations representing a range of academic disciplines. The final list included several patterns: verb plus noun (e.g. cast doubt), adjective plus noun (e.g. academic writing), noun plus noun (e.g. background knowledge), verb plus adjective (e.g. make explicit) and adverb plus adjective (e.g. highly controversial). A closer look at the distribution of collocations revealed that the greatest number of these patterns were made of nominal collocations, whereas adverb+adjective patterns formed the smallest category. Liu (2012) compiled a similar list incorporating multiword constructions, another name for lexical bundles, from previous research and looked at their distribution in two large corpora: the Corpus of Contemporary American English (COCA) and the British National Corpus (BNC). The master list includes a total of 228 bundles, most of which are phrasal and tend to be domain-specific. Functionally, patterns fall into one of three categories: referentials (e.g. in terms of the), stance markers (e.g. we argue that the) and discourse organizers (e.g. as a result of). The Phrasal Expressions List, compiled by Martinez and Schmitt (2010), is another attempt to generate a list of recurrent patterns for pedagogical use. By applying several criteria, such as frequency of occurrence, meaningfulness and non-compositionality, authors aggregated a total of 505 phraseological patterns derived from the British National Corpus. A final, and by far the most methodologically robust, attempt to suggest a list of academic bundles was conducted by Simpson-Vlach and Ellis (2010), who applied a combination of corpus-based and human intervention criteria for identifying potential bundles in academic speech and writing. Instead of reporting each of these characteristics separately, the authors adopted an innovative way in which bundles were arranged according to a composite score-Formulaic Worthy of Teaching (FWT). While the list is compelling, thus incorporating patterns of great utility for learners, it leans heavily toward general academic use, not discipline-specific one.

While all these lists are of greater pedagogical benefit which language instructors can draw upon in teaching as well as in preparing teaching materials, it is important to note that they are not without problems. First, and with the exception of the list generated by Wood and Appel (2014), the focus of these lists is on several domains and registers, a procedure that carries the risk of sidelining some core, discipline-based recurrent patterns typical of specific academic domains. A second problem relates to the concern raised by Hyland and Tse (2007) who questioned the merits of creating a one-size-fits-all list that could be used irrespective of the target domain or discipline. Third, the items in these lists are not comprehensive enough to account for recurrent patterns in some underexplored disciplines such as mathematics. This study thus aimed to address this vacuum by exploring a corpus of textbook materials for patterns of repeated vocabulary use. To direct the process of this research, answers to the following questions are sought:

1 The construct of pedagogical utility

Given that the key purpose underlying this study is to generate a list of pedagogically useful vocabulary for language-learning purposes, it is important to select items on the list based on sound methodological considerations, as is summarized below:

2 Study corpus

The corpus used in this study consists of 36 full-length textbooks from a series of publications aimed for graduate students in mathematics (for the full list of texts, see Appendix 1). The texts cover a wide range of mathematical topics, including enumeration, algebra, probability, calculus, number theory and matrices, to mention but a few. In selecting these textbooks, no decision is taken to differentiate between pure and applied mathematics, as a distinction of this type ‘is often difficult to make and lacks a clearly argued epistemological basis’ (Graves et al., 2014, p. 2). Prior to corpus treatment, texts are cleared of acknowledgements, tables of contents, lists of references, and indexes. Unlike in Wood and Appel (2014), activities and exercises sections, whether embedded in the text or supplied at the end of chapters or sections, are not removed, given their status as part of the text. Each textbook is given a distinct code so as to facilitate concordance checks needed to refine data and determine the functions of lexical patterns. Textbooks are then converted into plain texts, allowing for their treatment using corpus-automated tools. The Cluster Function in the WordSmith program (Scott, 2016) is used to extract recurrent patterns meeting the predetermined length and frequency scores.

3 Sequence identification

The identification process necessitates determining the number of words included within a single bundle, the frequency of the target bundles and their distribution across the texts making up the corpus. As for the length of the target sequence, there appears to be a tendency in lexical bundles research to target patterns incorporating four words (Cortes, 2004). With respect to the frequency of occurrence, there is no consensus among scholars on a specific frequency cut-off score beyond which a specific bundle can be included for analysis. In the literature, frequency cut-off scores vary from one study to the other, ranging from 10 (Simpson-Vlach & Ellis, 2010) to 50 times per million words (Breeze, 2013). Given these two extreme frequency thresholds, a rather moderate frequency score of 25 times per million words was adopted. A third consideration in the selection process of bundles concerns the number of texts in which the target expression should occur before being included in the final list of target bundles. The impetus behind this procedure is to ascertain that the recurrent pattern is not the result of an idiosyncratic use typical of a particular author or text (Biber & Barbieri, 2007). Some researchers have adhered to a minimum number of texts (e.g. Biber et al., 2004), while others consider measuring distribution according to a specific proportion (e.g. Hyland, 2008b). Given that the goal underlying this study is to offer mathematicians and specialists access to a vocabulary list with greater utility, a lexical bundle is included for analysis if it is found in 75% of the texts making up the corpus. This conservative distribution score aims to ensure that bundles appearing in the list are in wide use by different authors and are not a characteristic of a specific text or writer.

By applying these criteria, I am aware of the fact that a lot of potentially pedagogically useful sequences will not be included in the final listing simply because they fall below the predetermined distribution or frequency thresholds. However, this list is built with a goal that the candidate sequences will be incorporated into language teaching programs or used for creating pedagogic materials and including a large number of them will overwhelm students and overburden instructors. The process of applying length, frequency and dispersion parameters has resulted in an initial list comprising a total of 98 bundles.

4 Data refinement

The list obtained after applying length, frequency and range criteria is not without problems. An extensive process of checking concordance lines has shown a great deal of overlapping between similar bundles. Including overlapped bundles in a list may not be appropriate, given that one purpose underlying this research is to provide practitioners and math educators with various types of academic bundles, not variants of the same bundle. An example of such overlapping is found in sequences such as and only if the, and only if it, and and only if there, which are extensions of the bundle if and only if. Following the practice in previous research (e.g. Chen & Baker, 2010), closely related bundle types are combined into a single bundle. Another step to filter the list involves merging bundles that are identical except for one word, normally occurring in the bundle-final position. The bundles that there is a, then there is a, and in this case the are similar to the sequences that there is an, then there is an and in this case we except for the item occurring at the end of the sequence. To account for the existence of a closely related bundle form, the different word is separated from the original item by a slash (e.g. that there is a/an). Another refinement procedure addresses a group of lexical bundles which begin with the directive cognitive verb let and differ in a single variable occurring in the same position (e.g. let x be a, let g be a). This pattern is seen as a productive frame (Cunningham, 2017) which allows for various variables to be inserted within its boundaries. There are five let-initiated patterns with different frequency profiles, so the decision was taken to include the most recurrent pattern with an asterisk above it, indicating that other variables of the same frame also exist. Given the nature of the corpus under study, the retrieval process may entail academic sequences that are made of more symbols than words. Following McGrath and Kuteeva (2012) bundles including mathematical symbols or characters denoting mathematical entities are not included in the data (e.g. a and b are). The list after refinement incorporates a total of 71 bundles which will be further filtered after obtaining the opinions of math experts.

5 Expert opinion

In order to select items worthy of classroom attention from the corpus-derived list, it was decided to seek the opinions of 20 professionals with experience in publishing scholarly papers and materials in mathematics. Ackermann and Chen (2013, p. 236) have warned that ‘only with human intervention can a data-driven collocation listing be of much pedagogical use while still taking advantage of statistical information to help identify and prioritize the corpus-derived items.’ Experts handling the ranking process are chosen based on four criteria: expertise in mathematics education (years of teaching mathematics), publication record (they must have published a textbook, textbook chapter or a scholarly article prior to this study), academic rank (at least an assistant professor), and language of publication (English as a medium of publication). The participants include nine full professors, eight associate professors and three assistant professors. Their average length of experience is 25.55 years (Minimum 6, Maximum 43, SD = 11.06), and the average number of academic papers they have published is 35.10 (Minimum 2, Maximum 130, SD = 36). Besides scholarly papers, two participants have reported publishing eight textbooks (four texts each) and a third participant, seven texts. All experts have mentioned that they use English for publication with one of them adding French. The refined list of bundles resulting from the corpus treatment was given to these experts who were requested to evaluate items based on a 5-point Likert scale, in response to whether they think the sequence merits classroom attention and thus should be included in a list for learning purposes. The survey is presented to them with a brief explanation of the study goals, filling-in procedures, and the full list of sequences. Their responses range from extremely agree to extremely disagree:

The intra-class correlation coefficient is 0.73, showing moderate agreement among raters. A bundle was removed from the list if the sum of the expert judgments on it was under 80. The impetus for adhering to such conservative score is to ensure that items in the final list are of great pedagogical utility. Expert ratings have reduced the number of sequences to 65, thus eliminating six bundles deemed unworthy of classroom attention.

Expert opinions were also sought when determining the functions of the target bundles. The functional analysis was conducted by four professionals, none of whom took part in the survey measuring the usefulness of bundles. Here is a brief description of each expert handling the functional classification:

A Cronbach alpha was conducted to measure inter-rater agreement, and it was found at .56. This moderate result is admittedly unsurprising, given the highly specialized nature of the math register. The panel decided to meet and, as a group, work out the discrepancies with the help of concordance lines. A unanimous agreement was reached and the bundles were functionally classified based on the group decision of the panel members.

1 Structures of bundles

The literature is replete with several frameworks developed to classify bundles according to distinct structural forms (e.g. Biber et al., 2004; Chen & Baker, 2010; Cortes, 2004; Hyland, 2008a; Pan et al., 2016). Following that tradition, all items in our list are assigned to three major groups: noun-, preposition- and verb-based bundles. A fourth group is created to account for fragments that do not fall neatly into any of the three structural categories. As can be seen in Table 1, the first group of bundles comprises noun phrases, the largest number of which include embedded -of constructions. Some nominal bundles are preceded by the definite article the such as the proof of theorem and the sum of the, while others take the indefinite article a or its variant an (e.g. a linear combination of and an example of a). For the sake of space, I will mention two examples illustrative of each structural type.

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Table 1.

Nominal lexical bundles.

Structural category	Grammatical pattern	Lexical bundle
NP-based	NP + of	1. the set of all 2. the proof of theorem 3. (as) + the proof of the 4. the definition of the 5. a finite number of 6. the existence of the/a 7. the sum of the 8. a linear combination of 9. an example of a 10. (is) + a consequence of the 11. a special case of
	Other NPs	1. the right hand side 2. the fact that the 3. the left hand side

The second category includes preposition-headed sequences, almost half of which end with an -of pattern (Table 2). The remaining set of sequences begin with a preposition and conclude with either a marker for the beginning of another clause (e.g. in this section we) or a post-modifier (e.g. in the case the/we). The following are examples of sequences in this group:

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Table 2.

Prepositional lexical bundles.

Structural category	Grammatical pattern	Lexical bundle
PP-based	PP + of	1. in the proof of2. in the case of3. in terms of the4. from the definition of5. in the definition of
	Other PPs	1. in this section we2. with respect to the3. in this chapter we4. from the fact that5. on the other hand6. in the next section7. in this case the/we8. in such a way

In contrast to the two nominal categories, the third structural type consists of constructions comprising a verb component. These patterns can be further subcategorized into active/passive verbs, copula-verb-based constructions, existential there, and it-clauses. Apparently, the use of the verb be appears to dominate this group, as it makes presence in several subcategories. The first group of verb-based sequences consists of copula-be verb followed by noun, adjective, or prepositional phrases:

As can be seen in Table 3, other verbal subgroups incorporate existential ‘there’, it-clauses, and active/passive constructions. Here are examples of each type:

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Table 3.

Verb-based lexical bundles.

Structural category	Grammatical pattern	Lexical bundle
VP-based	Copula be + NP/AP	1. be the set of2. is the set of3. is a set of4. is an element of5. is equivalent to the6. is the same as + (the)7. is equal to the8. is of the form9. is independent of the
	Existential phrase	1. that there is a/an2. there is a unique3. then there is a/an4. then there exists a5. that there exists a6. there exists a unique
	VP with active verb	1. we may assume that2. let (x)* be a3. to show that the4. show that there is5. we see that the6. we have the following7. we say that a8. (it) + suffices to show that
	It-clauses	1. it is easy to + (to see that)2. it is enough to3. it follows from the4. it follows that the
	Passive verb + PP fragment	1. is said to be2. can be written as3. is contained in the4. is given by the5. can be used to6. can be found in

Table 4 includes the rest of the bundle structures which are mainly conditionals and fragments. Patterns with the conditional if are very frequent in the data, the most recurrent of which is the form if and only if:

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Table 4.

Other structural types of lexical bundles.

Structural category	Grammatical pattern	Lexical bundle
Other structures	Conditionals	1. if and only if + (the)/(it is)/(there)2. if there is a
	Fragments	1. such that for all2. and the fact that3. that the set of

2 Functions of the bundles

Another key purpose underlying the present study is to categorize bundles according to the functions that they serve in the texts. The functional framework developed by Hyland (2008a), which was subsequently applied by other researchers (e.g. Durrant, 2017; Pan et al., 2016), is chosen over that of Biber et. al (2004) because an initial piloting reveals that Hyland’s framework is comprehensive enough to account for the numerous discoursal functions exhibited by the sequences in the list. This is not surprising, given that the framework was first used to put groups sequences commonly found in academic writing into functional groups. According to this functional scheme, recurrent clusters are divided into three main groups: research-oriented, text-oriented and participant-oriented. Bundles serving a research-oriented function ‘help writers to structure their activities and experiences of the real world’ (Hyland, 2008a, p. 49). This functional group consists of several bundles that signal location, procedure, quantification, description, and topic. The second major category involves text-oriented bundles that are ‘concerned with the organization of the text and its meaning as a message or argument’ (2008a, p. 13). Bundles in this category are classified into transition, resultative, structuring, and framing signals. The third functional group includes participant-oriented bundles which serve to highlight the range of attitudes, opinions or judgements expressed by the writer. Bundles in this group fall into two major categories: stance and engagement.

The most bundles in the list perform a research-oriented function (see Appendix 2). This is not uncommon, given that one of the most important communicative purposes of graduate textbooks is to disseminate content knowledge to a presumably less-informed audience. Bundles in the research-oriented group can be further divided into subgroups, thus serving to mark existence/uniqueness, allude to location, refer to a topic-related item or procedure, define mathematically-oriented concepts, and describe an attribute or elaborate on a procedure or operation. Existence/uniqueness markers include eight bundles: that there is a/an, there is a unique, then there exists a, then there is a/an, that there exists a, there exists a unique, show that there is and the existence of the/a. These patterns are used to allude to some mathematically specific entities, such as a property, subset, function or form. Here are two examples illustrating existence markers:

A second research-oriented subgroup incorporates topic-related bundles that denote mathematical notions, concepts or processes. Sequences in this group are built around the term set: the set of all, be the set of, is the set of, is a set of, that the set of. A second group of topic-related bundles revolve around the notion of proof and is comprised of three patterns: the proof of theorem, in the proof of and (as) + the proof of the.

Yet a third subcategory comprises 10 expressions that define concepts, describe entities or elaborate on operations and processes. Patterns such as the definition of the, from the definition of, and in the definition of all serve to discuss a concept or explain an operation whereas patterns such as a linear combination of, is independent of the, is given by the, is of the form and is contained in the are used to describe an element or operation. The expression in such a way is used to elaborate on an ongoing process or operation. The following statements illustrate these two functional subgroups:

The fourth research-oriented group includes two bundles that serve to showcase a logical connection between two conditions. The most frequent and widely dispersed topic-related bundle in the list is the bi-directional conditional if and only if. A similar, though less frequent, sequence is the bundle if there is a, which highlights an if–then logical operation. Here are two examples of both types:

A fifth research-oriented subgroup consists of location bundles that are used to indicate the place of an element or entity. Two related bundles represent this functional subgroup, namely the right hand side and the left hand side which are used to refer to the two sides of an equation.

The final research group incorporates bundles that are utilized for quantifying elements. Quantification is a common procedure in mathematics, and is represented by four bundles: such that for all, the sum of the, is an element of and a finite number of.

Turning now to text-oriented bundles, it appears that lexical bundles are used to serve several sub-functions, including framing attributes, comparing and contrasting entities, directing the reader’s attention to parts of the text, announcing results and showing transition. Framing bundles include with respect to the, in the case of, in terms of the, a special case of, an example of a and in this case the/we. Due to space limitations, the following statements are examples of a selected number of bundles.

Some text-oriented bundles are used to draw contrasts and comparisons between elements and entities. The comparison/contrast sub-category contains four bundles: on the other hand, is equivalent to the, is the same a + (the) and is equal to the.

Another text-oriented sub-group consists of bundles that serve to structure the text (in this section we, in this chapter we, we have the following, in the next section, it follows from the, it follows that the). Here are some examples representing the two types:

Two text-oriented bundles are resultative markers, helping to lay out a mathematical outcome or to announce the result of a math operation. Bundles serving this function include is said to be and (is) a consequence of the.

The final, and by comparison the smallest, functional group incorporates bundles that perform a participant-oriented function. Lexical bundles in the first participant-oriented subgroup help to emphasize the centrality of the arguments laid out by mathematician writers. This subgroup is represented by three bundles, namely the fact that the, from the fact that and and the fact that, which revolve around the epistemic noun fact.

Bundles forming the third group include engagement markers that are initiated by the first person plural pronoun we. The engagement group has three bundles: we may assume that, we see that the and we say that a. These sequences can be exemplified by the corpus-derived statements below:

A similar subgroup of participant-oriented bundles include incomplete clauses that are headed by the anticipatory it and serve as attitude markers. More specifically, they function to reflect either the ease of processing, as in the sequence it is easy to (to see that), or sufficiency, as in the patterns it is enough to and (it) suffices to show that. Examples of these sequences are given below:

Two final participant-oriented bundles are used to appeal to the reader. The sequence to show that the tells the readers about a procedure undertaken to prove a theorem. The verb show here is used to mean prove. The sequence let (x) be a is utilized to alert the reader to the introduction of a new series of procedures, such as proofing a theorem:

It is important to point out here that some bundles tend to serve more than one function. Examples include patterns such as a special case of which may be interpreted from a stance point of view. By checking concordance lines, however, it seems obvious that this bundle is predominantly used as a framing device, thus serving a text-oriented function. The expression if and only if can be considered as a text-structuring bundle. After discussing this bundle with one informant, it becomes clear that this pattern is inextricably linked to the mathematical domain and would better be labeled as a research-oriented bundle.