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The Language of Mathematics:
A Corpus-based Analysis of Research Article Writing
in a Neglected Field
Laurence Anthony and Mark Bowen
Waseda University, Japan
Biodata
Laurence ANTHONY is Professor of Educational Technology and Applied Linguistics at
Waseda University, Tokyo, Japan. His main interests include tools development for corpus
linguistics research and applications, and English for Specific Purposes (ESP) program design
and methodologies. He is the developer of several corpus tools including AntConc,
AntWordProfiler, and AntMover. E-mail: anthony@waseda.jp
Mark BOWEN is Associate Professor of Mathematics at Waseda University, Tokyo, Japan. His
main research interests are high-order nonlinear partial differential equations, and exploiting a
combination of analytical and numerical techniques in their study. He is also interested in
developing teaching approaches and materials for native and non-native speakers of English in
the sciences and mathematics. E-mail: mbowen@aoni.waseda.jp
Abstract
Research article writing has received a great deal of attention from ESP researchers. Analyses of
the general structure of Introduction-Method-Results-Discussion (IMRD) articles, as well as
detailed analyses of individual sections, including introductions, results, and discussions
sections, have dominated the ESP literature, especially following Swales' pioneering work on
introductions in the early 1980s. Surprisingly, however, the writing of mathematics research
articles has been almost completely neglected to date. A few reasons for this can be speculated,
including a) the assumption that mathematics writing is similar to that in the well-covered areas
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of science and engineering, b) the difficulty in analyzing mathematics research articles due to
their often extremely specialized content, and c) the difficulty in locating expert mathematicians
who would be willing to serve as specialist informants. In this paper, we present an overview of
mathematics writing based on a corpus-based analysis of 410 refereed journal articles covering
one complete year of publications in a high-impact mathematics journal. The two-million word
corpus was divided into sections, and then analyzed using various corpus tools. Next, the
analysis was interpreted by the authors, both of whom have a background in mathematics and
one of whom is an active and well-published researcher in mathematics. Results of the study
reveal that some macro-level aspects of mathematics writing, such as the basic structuring of
titles and introductions, can resemble writing in the fields of science and engineering. On the
other hand, many features of mathematics writing diverge greatly from the established norms.
We offer reasons for these differences and suggest strategies for teaching writing to a mixed
group of science and engineers that may include mathematics majors.
Keywords: ESP, mathematics, research article writing, corpus-based analysis
1. Introduction
Research article writing has received a great deal of attention by ESP researchers, especially in
the 1990s and early 2000s following Swales' (1981) seminal study of research article
introductions (Swales, 1981) and his follow up study that introduced the CARS model (Swales,
1990). To date, studies of research article writing have focused almost exclusively on articles
that follow a standard Introduction-Methods-Results-Discussion (IMRD) structure, which has
been described extensively in the literature on report writing (e.g., Day, 1979; Swales & Feak,
2004; Robinson et al., 2008). A review of journal articles published in English for Specific
Purposes, for example, reveals two articles on research article titles, eleven on introductions, one
on the methods section, four on the results section, three on discussions, and one covering the
conclusion section. The interest in research article writing is highlighted further by noting that
five of the top ten most cited papers listed in 2012 in English for Specific Purposes are related to
research article writing (e.g., Ozturk 2007, Matsuda & Tardy 2007).
Despite the strong interest in research article writing among ESP researchers, there has
been a huge variation in the extent to which the genre has been investigated across different
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fields and disciplines. Figure 1 shows the results of a keyword search for various field and
discipline names in the titles of published articles in three of the top international ESP journals,
i.e., English for Specific Purposes, Asian ESP Journal, and ESP World. Clearly, research on
business, science, and medical English has dominated the ESP literature. Also, studies on legal
English, engineering English, and economics English (possibly a sub-category of business) have
been featured although less prominently. What is interesting to observe from Figure 1, however,
is the neglect of another major field of study, i.e., Mathematics. To date, only one research paper
focusing specifically on mathematics has been published in English for Specific Purposes, and
no papers have been featured in the other two journals under study.
Figure 1: Search terms appearing in titles of research articles in three ESP Journal
It is difficult to ascertain the reasons why mathematics has been neglected as an area of
ESP research to date. However, McGrath & Kuteeva (2012), the authors of the only published
ESP paper on mathematics, offer three possibilities. First, they argue that researchers may
assume that the language of mathematics is similar to that of hard sciences, such as physics,
chemistry, and biology, or theoretical disciplines, such as astrophysics, biostatistics and
theoretical physics. Indeed, Swales & Feak (2004) follow the latter view. Second, it is possible
that ESP researchers consider mathematics to deviate too much from the 'norm' of other sciences
to be included in a research study on cross discipline differences, such as that by Hyland (2006).
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Third, McGrath & Kuteeva suggest that researchers may consider mathematics discourse to be a
type of standardized code that requires little analysis. There is also a fourth possibility. To do a
detailed study of mathematics writing inevitably requires a specialist informant from the target
field. However, a myth continues that mathematicians are misunderstood, antisocial loners
(Devlin 1996). Therefore, ESP researchers may consider that finding an expert mathematician
that is willing to serve as a specialist informant is difficult. In fact, the opposite is generally the
case. Mathematicians are increasingly providing specialist knowledge of mathematics as part of
cross-discipline teams in order to solve complex real-world and theoretical problems. The point
is highlighted by the fact that this study is also a cross-discipline endeavor with specialist
knowledge of mathematics provided by a practicing mathematician.
Although the study of mathematics English has been largely ignored in the ESP literature,
it has featured in other areas of research. For example, there has been a great deal of research that
looks at how mathematics is taught in the classroom (e.g., Rowland 1995, 1999; Artemeva &
Fox 2011; Street 2005; Leung 2005; Barwell 2005; Morgan 2005; Pimm 1984). Researchers
have also looked at how language (not English per se) is used to explain mathematical concepts
(e.g., Huang & Normadia 2007; Borasi & Rose 1989; Connolly & Vilardi 1989; Buerk 1990;
MacGregor 1990; Countryman 1992; Johanning 2000). In the area of English language, some
rare studies on mathematics language can be found, such as an analysis of imperatives (Swales et
al. 1988), language and symbolism (O'Halloran 2005), authorial identity (Burton & Morgan
2000), and the above study by McGrath & Kuteeva (2012) on stance and engagement. Clearly,
educationalists are interested in the language of mathematics and would find great value in the
results of ESP studies on mathematics. Not only would the results help to improve classroom
teaching of mathematics, they would also help to build better metacognitive views of
mathematics, and of course, help ESP teachers meet the needs of mathematicians in academic
and technical reading/writing classes.
In this paper, we report on a study of the writing of research articles in mathematics at
both the macro and micro level. At the macro level, we investigate the presence and positioning
of the major sections of the research article and compare the results with those for a more
traditional science/engineering field, namely mechanical engineering. As a result of this analysis,
we hope to establish to what extent mathematics writing (and indeed mechanical engineering
writing) follows an IMRD structure.
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At the micro level, we look at the writing style of mathematics research articles and
identify in what ways the style of mathematics writing resembles or differs from that in a more
traditional science/engineering field science and engineering (again, mechanical engineering).
Previous literature on research article writing in science and engineering (e.g. Robinson et al.
(2008), Sales (2006) and Swales & Feak (2004)) has suggested that a formal style in
predominantly used by authors. A formal style manifests itself in many ways, including the
infrequent use of imprecise, general conversation words and expressions such as "stuff,"
"things," "bunch," and "whole lot of," the infrequent use of phrasal verbs (e.g. "figure out," "
make up", "go down"), and the infrequent use of connectives such "and," "so," and "but" that
even tools such as Microsoft Word mark as being informal (Swales & Feak, 2004: 17-24). Here,
we investigate the features of these three styles in the hope of establishing whether mathematics
writing adopts a formal or informal style. We address the following two research questions:
1. Does mathematics research article writing diverge from the 'norm' of science and engineering
research article writing in terms of macro-level structuring, i.e., the presence and positioning
of the IMRD ("Title", "Abstract," "Introduction," "Methods," "Results," and "Discussion")
sections? If yes, in what way does it diverge from the 'norm'?
2. Does mathematics research article writing diverge from the 'norm' of science and engineering
research article writing in terms of style (i.e. does it deviate from a formal writing style)? If
yes, in what way does it diverge from the 'norm'?
For the analysis, we use a large corpus of 410 refereed journal articles comprising one
complete year of published works in a high-impact mathematics journal. First, we create a
structural model of mathematics research article writing based on a qualitative analysis of the
texts by the two authors, both of whom have degrees in mathematics and one of whom is an
active and well-published researcher in the field. Next, we conduct a quantitative corpus-based
analysis of the texts to confirm or reject the structural model and to identify characteristic
features of style. We then compare these corpus-based results with those of a comparable
reference corpus of texts from a more traditional science/engineering field. Finally, we discuss
the results and offer implications for ESP teaching in mathematics.
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2. Methodology
2.1 Corpus design
For a corpus-based analysis of writing, it is necessary for the target corpus to be both balanced
and representative of the target language (Biber, 1993). It is also important that the corpus has
extrinsic validity, i.e., users of the results of the study can understand the relevance and
applicability of the results. To ensure a reasonably balanced selection of mathematics topics, an
ideal corpus would contain articles from a wide range of research journals covering both pure
and applied mathematics. Also, to ensure that each journal was represented accurately in the
corpus, a large number of articles from each journal would need to be selected. However,
maintaining balance and representativeness would necessitate including many articles from less-
prestigious journals. Thus, the validity of the study may be questioned, especially considering
that the aim of the study is to provide useful results to teachers of ESP.
For this study, we chose to relegate the importance of balance and focus instead on
creating a representative and valid corpus comprised of all articles published in one year of a
single, high-impact mathematics journal. To ensure that the corpus was valid, we employed the
following journal selection criteria:
The journal should be ranked in the top 10 highest impact factor journals in the area of
applied mathematics, according to the Thomson Reuters (formerly ISI) Web of
Knowledge (http://wokinfo.com/).
The journal should not be a review article journal.
The journal should cover a broad range of mathematics domains.
The journal should appeal to a broad audience of both pure and applied mathematicians.
Based on the above criteria, we selected Nonlinear Analysis: Real World Applications
(hereafter NARWA) for the analysis and collected all 410 articles (approx. 1.9 million words)
from Volume 11 (year 2010) of the journal. Details of the target corpus are given in Table 1.
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Table 1: Target corpus details
JOURNAL TITLE: Nonlinear Analysis: Real World Applications
(NARWA)
PUBLISHER: Elsevier
IMPACT FACTOR (2012): 2.043
DATES: February 2010 - December 2010
(Volume 11: Issues 1-6)
SAMPLING: Whole population approach
(410 articles: 1 entire year)
CORPUS SIZE: 1,917,422 tokens; 30,700 types
It was also necessary to create a corpus of research articles from a more traditional
science/engineering field to serve as a comparison. For this study, we chose to analyze articles
from the field of mechanical engineering. This is a traditional applied engineering field, which in
some ways can be considered to be the exact opposite of mathematics. The following selection
criteria were employed:
The journal should be ranked in the top 10 highest impact factor journals in the area of
mechanical engineering, according to the Thomson Reuters (formerly ISI) Web of
Knowledge (http://wokinfo.com/).
The journal should not be a review article journal.
The journal should cover a broad range of mechanical engineering domains.
The journal should appeal to a broad audience of mechanical engineers.
Based on the above criteria, we selected the Journal of Engineering Materials and
Technology (JEMT) for the analysis and collected all 318 articles (approx. 1.6 million words)
from Volume 122 (year 2000) of the journal. Details of the target corpus are given in Table 2.
Table 2: Target corpus details
JOURNAL TITLE: Journal of Engineering Materials and Technology
(JEMT)
PUBLISHER: American Society of Mechanical Engineers (ASME)
IMPACT FACTOR (2012): 0.56
DATES: January 2000 – December 2000
(Volume 122: Issues 1-4)
SAMPLING: Whole population approach
(318 articles: 1 entire year)
CORPUS SIZE: 1,643,576 tokens; 24,637 types
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2.2 Software tools
The analysis of the corpus data was carried out using the AntConc 3.3.5 concordancer analysis
toolkit (Anthony, 2012), and specially written Python scripts developed by one of the authors
(Anthony).
2.3 Qualitative analysis
In order to create an intuitive model of mathematical papers that was not influenced by the
corpus data or later quantitative analysis, we (the authors) first discussed at length our own
experiences of writing and reviewing mathematics papers. Next, we formulated a preliminary
model and then critiqued this model, clarifying, simplifying, or expanding each of its steps to
arrive at a final model that the results of the quantitative analysis could be compared against. On
completion of the quantitative analysis, we also reviewed the results and discussed the
interpretations and implications of the findings with a view to formulating recommendations for
future ESP teaching in mathematics.
3. Results
3.1 Macro-level structure of mathematics papers – Intuitive model
Our intuitive model of mathematical research articles is shown in Figure 2. First, it is important
to note that mathematics papers can generally be classified on a cline between analytical papers
(pure mathematics) and application papers (applied mathematics). We considered the general
structure of the two to be equivalent, i.e., they would both include a title, an abstract, and an
introduction, and they would discuss the background to the research, explain the methods and
results, and optionally finish with some kind of discussion/conclusion section. However, we also
agreed that the purpose and details included in each section would reflect the nature of the study
and thus vary greatly. For example, in an analytical paper, the purpose is often to prove some
mathematical result (stated in the background) and thus the result 'section' would simply offer a
closure to the study and may consist of just a single sentence. On the other hand, in an
application paper, the aim of the research would be to propose a number of mathematical
relationships relating to some real-world applications. Thus, the results of the study could be of
three different types: a) details of the relationships, b) experimental results that are used to
validate the mathematical relationships in a real-world setting, and/or c) computer simulations
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that verify the validity of the relationships and perhaps offer further insights into the
relationships. Also, we anticipated that the nature of the study would also dictate the actual
section headings used in the research articles. We anticipated few would use "Background,"
"Methods," "Results," or "Discussion/Conclusion" as heading labels.
Figure 2: Intuitive model of mathematics research article structure
3.2 Macro-level structure of mathematics papers – Corpus-based model
To confirm or reject the intuitive model presented in section 3.1, we first wrote a Python script
that would extract and count the number of section headings from each article in the target and
reference corpora. Figures 3 and 4 show the results of the analysis.
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Figure 3: Percentage Occurrence of Section Headings in NARWA
Figure 4: Percentage Occurrence of Section Headings in JEMT
Figures 3 and 4 show the percentage occurrence of different labels in the mathematics
(NARWA) and mechanical engineering (JEMT) articles, respectively. In both figures, individual
headings in the articles are given an identification (ID) number and arranged in order from the
most frequent to the least frequent one. In both figures, the most frequent heading (ID = 1) is
"Introduction". Figures 3 and 4 highlight that research articles in both mathematics (NARWA)
and mechanical engineering (JEMT) show a great variation in the headings used. In NARWA,
951 different main headings were used (ave. 6.0 per article) with 832 (87%) of them occurring
just once. Examples of single occurrence headings are "Exponential convergence," "Blow up
phenomenon," and "Classical global solutions." In JEMT, 1167 different headings were used
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(ave. 7.1 per article) with 1081 (93%) of them occurring just once. Examples of single
occurrence headings are "Density measurement," "Previous work," and "Thermal and
mechanical analyses."
Figure 5: Percentage Occurrence of Title, Abstract and Top 10 Sections in NARWA
Figure 6: Percentage Occurrence of Title, Abstract and Top 10 Sections in JEMT
Figures 5 and 6 show the frequency of occurrence of the title, abstract, and top ten most
commonly used section headings in the NARWA and JEMT corpora. Not surprisingly, most
articles in both corpora include an introduction and references. Just over half the articles in both
corpora have acknowledgments. However, beyond these very typical sections, few other
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generalizations can be made. Clearly, mathematics articles do not typically include "Methods,"
"Results," or "Discussion" sections; a fact that was predicted from the intuitive model. However,
surprising to these authors was the fact that mechanical engineering papers also did not show the
typical IMRD pattern.
3.3 N-gram analysis of NARWA section titles
The results in Figures 3 and 5 show that mathematics research articles in the NARWA corpus do
not show a general IMRD labeling of section headings. This was predicted by the intuitive model
given in Figure 2. However, in order to confirm or reject the basic structuring of mathematics
article content predicted by the model, it was necessary to clarify whether the variously named
sections of the NARWA corpus articles showed any general patterns in terms of content, and if
so, establish what these were.
In view of the fact that most section headings occurred only once in the corpus, a
complete analysis would require all the articles to be read in full and subsequently analyzed for
content structuring. Not only would this be extremely time consuming, it would also introduce
the danger of researcher bias, i.e., seeing patterns in the data that did not exist. To counter this
danger, we decided to adopt a quantitative analysis of the section structuring based on counts of
N-grams (contiguous word sequences of length N) of varying lengths in the different sections.
We could then relegate the qualitative analysis to only interpretations of the N-gram frequency
tables.
To carry out the analysis, first, the section headings of each article in the NARWA corpus
were grouped according to their order of appearance in the article as a whole. In this way, the
opening sections of the articles, with headings such as "Introduction," "Introduction and
Preliminaries," and "Introduction and Main Results" were grouped together. Similarly, the
headings used for the second sections of the articles were grouped together and so on until all
headings were accounted for. Next, common patterns of structuring in each group were
established by counting N-grams of size one to six using the AntConc 3.3.5 (Anthony, 2012) N-
gram tool, and ranking them according to frequency. The maximum size of N-gram was chosen
based on the frequency distribution of section headings in the corpus, where 374 (91%) of the
articles contained six or less sections and just 36 (9%) of corpus articles included seven sections
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or more. Following this approach, the highest ranked N-grams would represent the most
commonly used patterns in each section.
Due to space restrictions, Table 3 shows the results of the analysis for N-grams of size
one to three for the first six sections of the corpus articles. Raw frequencies are given for each N-
gram in the column adjacent to the N-gram. The N-gram results show several very commonly
used headings, For example, Section 1 of the articles reveals a very strong preference for the
single word heading "Introduction" although minor variations are possible, such as "Introduction
and Preliminaries".
Table 3: Top 10 N-Grams of size 1-3 for each section of the NARWA corpus.
Section 1
introduction 401 introduction and 11 introduction and main 7
and 13 and main 7 and main results 4
main 7 and preliminaries 4 and main result 3
preliminaries 5 main results 4 introduction and preliminaries 3
results 4 main result 3 a kinetic model 1
result 3 and background 2 basic physical concepts 1
the 3 the problem 2 introduction and background 1
background 2 a kinetic 1 introduction notations and 1
model 2 basic physical 1 kinetic model q 1
problem 2 introduction notations 1 notations and background 1
Section 2
the 110 of the 42 of the problem 14
of 108 existence of 17 formulation of the 10
and 78 the model 17 existence and uniqueness 6
preliminaries 76 main results 15 the mathml source 6
#
model 65 the problem 14 view the mathml 6
#
formulation 42 preliminary results 13 of the model 5
problem 35 problem formulation 11 positive periodic solutions 5
results 31 formulation of 10 and existence of 4
existence 28 governing equations 10 and uniqueness of 4
mathematical 27 mathematical model 10 existence of hopf 4
Section 3
of 141 of the 49 of the main 8
the 113 main results 32 proof of the 7
and 68 existence of 22 existence and uniqueness 6
main 45 stability of 16 the existence of 6
results 44 proof of 15 analysis of the 5
existence 40 periodic solutions 11 existence of the 5
stability 39 main result 10 of the problem 5
solutions 33 existence and 9 solution of the 5
solution 28 hopf bifurcation 9 a priori estimates 4
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analysis 26 the existence 9 and uniqueness of 4
Section 4
of 95 of the 30 and uniqueness of 5
the 65 proof of 16 existence and uniqueness 5
and 48 numerical simulations 12 analysis and discussion 3
numerical 43 existence of 10 and stability of 3
results 22 and discussion 9 convergence of the 3
stability 21 numerical results 8 direction and stability 3
analysis 20 periodic solutions 7 existence of the 3
discussion 19 stability of 7 numerical results and 3
conclusions 18 existence and 6 of limit cycle 3
existence 17 analysis and 5 of the solution 3
Section 5
of 52 of the 15 results and discussion 6
conclusions 36 and discussion 8 an illustrative example 3
the 36 numerical simulations 8 asymptotic behavior of 2
and 33 concluding remarks 7 behavior of the 2
numerical 28 proof of 7 discussion and conclusions 2
conclusion 26 results and 7 nonconstant positive solution 2
discussion 23 numerical results 6 nonexistence of nonconstant 2
results 18 discussion and 5 numerical results and 2
simulations 11 illustrative example 4 of nonconstant positive 2
a 10 numerical simulation 4 proof of the 2
Section 6
conclusion 21 concluding remarks 7 results and discussion 3
of 21 of the 7 existence of nonconstant 2
the 17 numerical results 4 nonconstant positive solutions 2
discussion 14 results and 4 of nonconstant positive 2
conclusions 13 and discussion 3 of the positive 2
and 9 numerical examples 3 stability of the 2
numerical 9 proof of 3 the existence of 2
remarks 9 existence of 2 a convective condition 1
concluding 7 final remarks 2 a critical point 1
results 7 nonconstant positive 2 a heat flux 1
*Numbers in the adjacent columns are the raw frequency counts. N-grams in italics are likely to reflect sections from analytical papers. N-grams
in bold are likely to reflect sections from application papers. The classifications of the N-grams were determined by the subjective judgments of
this paper's authors based on their experience reading and writing mathematics papers.
#These entries relate to the markup of equations in the body of the text. They should be considered as noise.
However, the main result from Table 3 is that the majority of sections in mathematics
papers show a complex pattern of section structuring. To understand these patterns more clearly,
we went through the N-gram list and attempted to classify each N-gram into one of three
categories; a) those that are likely to relate to analytical work (shown in italics in Table 3), b)
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those that are likely to relate to applications (shown in bold in Table 3), and c) those that are
generic in nature (unmarked in Table 3). A summary of this analysis is shown in Table 4 with
characteristic N-gram patterns for each type of article appearing below a general description of
the section purpose.
Table 4: Summary of N-gram categorization of NARWA section headings.
Section 1 (introduction)
Analytical Papers Application Papers
introduction
introduction and main results
introduction and preliminaries
Section 2 (background/methods)
Analytical Papers Application Papers
main results
preliminary results
the model
mathematical model
problem formulation
model formulation
governing equations
mathematical formulations
Section 3 (methods/results)
Analytical Papers Application Papers
proof of the main result
solution of the problem
main result and its applications
linear stability and Hopf bifurcation analysis
direction and stability of the Hopf bifurcation
existence and uniqueness of theorem/solutions
analysis of the data/model/problem
a priori estimates of positive solutions
existence of a positive periodic solution
the main result/results
Section 4 (results/application)
Analytical Papers Application Papers
existence and uniqueness of equilibrium point
existence and uniqueness of limit cycle
direction and stability of the Hopf bifurcation
convergence of the series solutions
solution and equilibrium points
application
equation for the pressure
control design for the RTAC system
results and discussion
Section 5 (results/application)
Analytical Papers Application Papers
proof of the principal theorem
proof of the main results
nonexistence of nonconstant positive solution
periodic solutions
numerical results and discussion
numerical results
numerical simulations
simulation results
experimental results
results and discussion
illustrative example
an example
Section 6 (conclusions)
Analytical Papers Application Papers
concluding remarks
final remarks
concluding remarks
final remarks
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results and discussion
proof of ...
results and discussion
numerical examples
numerical results
The analysis in Table 4 is largely consistent with the intuitive model presented in Figure
2. Clearly, there is a general ordering of information in terms of introduction, background,
methods, results, and discussion/conclusion. However, this ordering may not be immediately
apparent unless the reader is well trained in the theories and practices of mathematics. Also, it is
clear from Table 4 that the choice of section structuring differs greatly depending on whether the
mathematics paper has a focus on analytical methods or applications.
3.4 Comparison of writing styles in the NARWA and JEMT corpora
A preliminary analysis of the NARWA corpus articles revealed multiple occurrences of
imprecise, general conversation words and expressions, phrasal verbs, and the connectives "and,"
"so," and "but," that were described by Swales & Feak (2004) as indicative of informal language.
Several examples are shown below, where the informal word or expression is highlighted in
bold.
It is easy to verify that U and F satisfy the operator equation.
There you can see the precise conditions...
Here, you note that the condition (%%%) above is satisfied.
We say that, the problem (%%%) and (%%%) is maximal regular...
Anyway, we have that EQT is bounded ...
It should be pointed out that discrete-time neural networks become more important...
Similarly the second-order solution works out to be EQT...
It turns out that depending on the locations...
So the stability of neural networks has been one of the most active areas of research.
But for the Lotka–Volterra predator–prey systems, it is more difficult to discuss.
And we arrive at the purpose of the present article.
In mathematics writing, the researcher is often taking the reader on a journey through
various theorems and lemmas to arrive at a proof or new model. In this exposition, a commonly
held view among mathematicians is that formality can be sacrificed in exchange for clarity
(Halmos, 1973). To investigate if this phenomenon is unique to mathematics writing, we looked
in both the NARWA and JEMT corpora at the frequency of occurrence of various imprecise
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words and expressions, the frequency of occurrence of phrasal verbs, and the frequency of
occurrence of the informal connectives "and," "so," and "but." The results are shown in Table 5.
Table 5: Frequency of occurrence of informal features in NARWA and JEMT
Informal Language Feature % Occurrence
NARWA JEMT
use of adjective "easy" 552 hits (0.46% of all adj.) 52 hits (0.04% of all adj.)
phrasal verbs 4697 hits (18% of all
verbs)
7776 hits (16% of all
verbs)
informal connectives 714 hits (7.6% of all
connectives)
and (155 hits: 1.6%)
so (372 hits: 4.0%)
but (187 hits: 1.9%)
182 hits (2.4% of all
connectives)
and (91 hits: 0.61%)
so (48 hits: 0.97%)
but (43 hits: 0.84%)
The results in Table 5 suggest that mathematics articles do indeed show a greater
tendency to use informal expressions than articles from mechanical engineering. In particular,
the adjective "easy" was used almost ten times as often in NARWA than in JEMT, and the word
"so" was used four times as often in the mathematics corpus. However, all the informal
expressions investigated also appeared in the mechanical engineering corpus. For this study, we
did not calculate if the differences in occurrence of informal expressions were significant.
However, there was a large variation in occurrence of informal expressions between different
articles, and so we anticipate that the differences are not significant.
4. Discussion
The first research question asked if mathematics research article writing diverges from the 'norm'
of science and engineering research article writing in terms of macro-level structuring. Although
our results revealed that mathematics articles are structured in widely varying forms and
consistently break the traditional IMRD model of Introduction-Methods-Results-Discussion, we
also discovered that this is also the case for mechanical engineering research papers. This was a
surprising result as we anticipated that mechanical engineering research papers would reveal a
more consistent pattern in view of its status as a well-established and traditional engineering
field. This result has profound implications for ESP teachers of writing in science and
engineering. Many textbooks focus on the IMRD structure of research papers. However, this
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assumed 'norm' of writing may be less 'normal' than previously assumed. In a real-world
scenario, rather than following the IMRD structure, students may be well advised to write their
research articles following a less rigid format. In mathematics, the best advice to give to students
may be to let the research determine the flow of the research article. For example, if the writer
anticipates that the reader will need some preliminary knowledge before understanding the
proposed model, then a preliminary knowledge section should be included. Similarly, if the
results naturally lead to some interesting applications, then the writer should feel able to include
an additional section describing these applications even if it comes between the results and the
discussion.
The second research question asked if mathematics research article writing diverged from
the 'norm' of science and engineering research article writing in terms of style. Again, the results
were surprising. Although expert researchers in mathematics did include informal expressions in
their writing, this phenomenon was not unique to their field. In fact, the same informal
expressions were observed in mechanical engineering writing, although to a lesser extent.
Writers in both mathematics and mechanical engineering used vague terms such as "easy", wrote
using phrasal verbs, and linked ideas together using the connectives "and," "but, and "so". All
these features are traditionally considered to be inappropriate for a formal academic writing style
(see Swales & Feak, 2004) and are even explicitly signaled as inappropriate by style checking
tools such as that in Microsoft Word. Clearly, ESP teachers need to be aware that informal
expressions can be used in advanced technical writing within some disciplines, and they need to
inform students of this fact in the writing classroom. Although it may be useful in beginner level
classes for teachers to encourage students to follow traditional models of writing style, as
students advance in their writing, perhaps a more relaxed view of style is necessary. One way to
achieve this is to expose students to corpora in the classroom and allow them to investigate
patterns in writing using a data-driven learning approach.
5. Conclusion
In this paper, we have investigated the structure of research article writing in the field of
mathematics and compared it to that in mechanical engineering. The results show that the
structuring of mathematics papers varies considerably from article to article and that few
consistent patterns in the choice of sectioning can be found. However, the same result was found
23
for mechanical engineering papers, suggesting that the traditional IMRD model is not as
prevalent in the research literature as it is depicted in textbooks on writing. At a deeper level,
however, mathematics papers do reflect the research process itself, starting with an introduction
and review of background work, continuing with a description of methods and results, and
finishing with a discussion or conclusion. This basic pattern is followed regardless of whether
the research is focused more on analytical methods or applications of mathematical models. In
terms of style, mathematics articles include various informal expressions that are often
considered inappropriate in writing textbooks. However, these same expressions were also found
to occur in mechanical engineering articles although to a lesser degree. Again, these findings
question the advice given in textbooks on writing and suggest that ESP teachers should
encourage learners to be more flexible in their writing and become more aware of actual patterns
of writing in professional journals in their field, perhaps through a data-driven learning approach
utilizing specialized corpora in the classroom.
Acknowledgements
The authors express their gratitude to two anonymous reviewers for comments on an earlier
version of this paper.
References
Anthony, L. (2012). AntConc (Version 3.3.5) [Computer Software]. Tokyo, Japan:
Waseda University. Available from http://www.antlab.sci.waseda.ac.jp/.
Artemeva, N.,& Fox, J. (2011). The writing’s on the board: The global and local in
teaching undergraduate mathematics through chalk talk. Written Communication,
28(4), 345–379.
Barwell, R. (2005). Language in the Mathematics classroom. Language and Education,
19(2), 96-101.
Biber, D. (1993). Representativeness in corpus design. Literary and Linguistic
Computing, 8(4), 243-57.
Borasi, R., & Rose B. (1989) Journal writing and mathematics instruction. Educational
Studies in Mathematics, 20, 347–65.
Buerk, D. (1990). Writing in mathematics: A vehicle for development and
24
empowerment. In A. Sterrett (Ed.), MAA Notes, Vol. 16. Using writing to teach
mathematics (pp. 78–84). Washington, D.C.: Mathematical Association of America.
Burton, L., & Morgan, C. (2000). Mathematicians Writing. Journal for Research in
Mathematics Education, 31, 429-453.
Connolly, P., & Vilardi T. (eds.) (1989). Writing to learn mathematics and science.
New York: Teachers College Press.
Countryman, J. (1992). Writing to learn mathematics: Strategies that work. Portsmouth,
NH: Heinemann.
Day, R. A. (1979). How to write, and publish a scientific paper. Westport, CT: Oryz.
Devlin, K. (1996). Of men, Mathematics, and myths. Devlin's Angle. Retrieved April 18,
2004 from http://www.maa.org/devlin/devlin_aug.html
Halmos, P. R. (1973). How to write mathematics. In N. E. Steenrod, P. R. Halmos, M.
M. Schiffer, & J. R. Dieudonne (Eds.), How to write mathematics (pp. 19-48) US:
American Mathematical Society.
Huang, H., & Normadia, B. (2007). Learning the language of mathematics: A study of
student writing. International Journal of Applied Linguistics, 17(3), 294-318.
Hyland, K. (2006). Disciplinary differences: Language variation in academic discourses.
In K. Hyland & M. Bondi (Eds.), Academic discourse across disciplines (pp. 17–45).
Frankfurt: Peter Lang.
Leung, C. (2005). Mathematical vocabulary: Fixers of knowledge or points of
exploration? Language and Education, 19(2), 126-134.
Johanning, D. I. (2000). An analysis of writing and postwriting group collaboration in
middle school pre-algebra. School Science and Mathematics, 100(3), 151–61.
MacGregor, M. (1990). Reading and writing in mathematics. In J. Bickmore-Brand
(Ed.), Language in mathematics (pp. 100–8). Portsmouth, NH: Heinemann.
Matsuda, P. K., & Tardy, C. M. (2008). Continuing the conversation about voice in
academic writing. English for Specific Purposes, 27(1), 100-105.
McGrath, L., & Kuteeva, M. (2012). Stance and engagement in pure mathematics
research articles: Linking discourse features to disciplinary practices. English for Specific
Purposes, 31(3), 161-173.
25
Morgan, C. (2005). Words, definitions and concepts in discourses of mathematics,
teaching and learning. Language and Education, 19(2): 103–17.
O'Halloran, K. L. (2005). Mathematical discourse: Language, symbolism and visual
images. London and New York: Continuum.
Ozturk, I. (2007). The textual organization of research article introductions in applied
linguistics: Variability within a single discipline. English for Specific Purposes, 26(1),
25-38.
Pimm, D. (1984). Who is ‘‘we’’? Mathematics Teaching, 107, 39–42.
Robinson, M. S., Stoller, F. L., Costanza-Robinson, & M. S., Jones, J. K. (2008). Write
like a chemist. Oxford, UK: Oxford University Press.
Rowland, T. (1995). Hedges in mathematics talk. Education Studies in Mathematics, 29,
327–353.
Rowland, T. (1999). Pronouns in mathematics talk: Power, vagueness and
generalisation. For the Learning of Mathematics, 19(2), 19–26.
Sales, H. E. (2006). Professional communication in Engineering. New York, US:
Palgrave.
Street, B. (2005) The hidden dimensions of mathematical language and literacy.
Language and Education, 19(2): 136–41.
Swales, J. M. (1981). Aspects of article introductions. Birmingham, U.K.: University of
Aston, Language Studies Unit.
Swales, J. M. (1990). Genre analysis: English in academic and research settings.
Cambridge, U.K.: Cambridge University Press.
Swales, J. M., Ahmed, U., Chang, Y., Chavez, D., Dressen, D., & Seymour, R. (1998).
Consider this: The role of imperatives in scholarly writing. Applied Linguistics, 19(1),
97–121.
Swales, J. M., & Feak, C. B. (2004). Academic writing for graduate students. Ann
Arbor, MI: Univ. Michigan Press.
