(PDF) Mathematics is the method: Exploring the macroorganizational structure of research articles in mathematics

Discourse Studies http://dis.sagepub.com/ Mathematics is the method: Exploring the macro-organizational structure of research articles in mathematics Heather Graves, Shahin Moghaddasi and Azirah Hashim Discourse Studies 2013 15: 421 DOI: 10.1177/1461445613482430 The online version of this article can be found at: http://dis.sagepub.com/content/15/4/421 Published by: http://www.sagepublications.com Additional services and information for Discourse Studies can be found at: Email Alerts: http://dis.sagepub.com/cgi/alerts Subscriptions: http://dis.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://dis.sagepub.com/content/15/4/421.refs.html >> Version of Record - Aug 14, 2013 What is This? Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 482430 2013 DIS15410.1177/1461445613482430Discourse StudiesGraves et al. Article Discourse Studies Mathematics is the method: 15(4) 421­–438 © The Author(s) 2013 Reprints and permissions: Exploring the macro- sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1461445613482430 organizational structure dis.sagepub.com of research articles in mathematics Heather Graves University of Alberta, Canada Shahin Moghaddasi University of Malaya, Malaysia Azirah Hashim University of Malaya, Malaysia Abstract This article reports the macro-organizational structure of research articles (RAs) in mathematics, based on an analysis of 30 published pure and applied mathematics articles. Math RAs eschew the Introduction-Methods-Results-Discussion (IMRD) structure for an Introduction-Results model that enables researchers to present new knowledge as clearly and succinctly as possible. Notable omissions from the mathematics RA structure are Method and Discussion sections, which mathematicians do not need because of the well-established methodology used in the field (based on deduction and induction) and the relative absence of extended discussion required to interpret research findings. We contextualize the macrostructure of RAs in mathematics within the discourse conventions and disciplinary assumptions about knowledge in the field to suggest the value of such a strategy to teachers and students of academic writing. Keywords Academic writing, argument in mathematics, genre analysis, macro-organization in research mathematics, structure of argument, writing in the disciplines Corresponding author: Heather Graves, Department of English and Film Studies, Humanities Centre 4-81, University of Alberta, Edmonton, Alberta, Canada T6G 2E5. Email:

Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 422 Discourse Studies 15(4) Introduction Even after 20 years, genre analysis of research articles (RAs) continues to be a generative area for new scholarship (see Appendix A). Most studies adopt Swales’s English for Specific Purposes (ESP) approach to genre analysis to determine the communicative purposes of text producers. The communicative purposes are catalogued as hierarchies of obligatory and optional Moves and Steps that suggest a model for a typical RA in that field, a model that is then recommended for pedagogical application by novice writers in the discipline. Genre study has tended in two directions, one focused on RA sections (that is, Introduction sections, Method sections, etc.) and the second focused on explaining the link between RA structure, methods used in the discipline and disciplinary epistemology. The first tendency produces partial knowledge of the genre structure in that field and limits the usefulness of the research from a pedagogical perspective. That is, teachers of academic discourse (e.g. English for Academic Purposes (EAP) or ESP) and academic writing course content developers need comprehensive knowledge of RA structure so they can help students learn what a well-written RA looks like in their particular disci- pline. Several studies have contributed to our comprehensive knowledge of RA structure in fields including medicine (Nwogu, 1997) computer science (Posteguillo, 1999), applied linguistics (Ruiying and Allison, 2004), and biochemistry (Kanoksilapatham, 2005). These studies are valuable resources for teachers tasked with teaching the applica- tion of knowledge from genre analysis. The second tendency, explaining links between article structure, methodology and disciplinary epistemology, is represented through studies by Brett (1994), Kanoksilapatham (2005), Parkinson (2011), Posteguillo (1999) and Samraj (2002), which make these connections and describe aspects of the RA. An essential contribution to genre analysis, these studies link move structure to disciplinary culture. In contrast, genre studies focused mainly on generating lists of Moves and Steps can imply that learning these typical argument structures equals acquiring genre knowledge. We do know that each discipline defines knowledge in slightly different ways so that new con- tributions to knowledge do not look the same in or to all disciplines. In addition, wide methodological variations shape how disciplines obtain and claim new knowledge. Therefore, understanding of Move structure must be situated within a student’s ethno- graphic awareness of how and what a particular discipline values as knowledge and research. Indeed, many scholars have emphasized the importance of a multidimensional methodological approach to genre study (Bhatia, 2004; Candlin and Hyland, 1999; Hyland, 2000; Swales, 1998). Based on these methodological contentions, we have analyzed the schematic structure of mathematics RAs based on a sample of 30 articles embedded within a larger analysis of the writers’ macro-framework and major sectioning strategies. Further, we describe, interpret and explore some of the argumentative moves used to present the findings within ethnographic accounts about the texts from four specialists in mathematics. Our analysis began with the macro-structure of RAs. In his overview of their macro-structure, Swales (1990) proposed an ‘hour-glass’ structure, established as the Introduction-Method-Results-Discussion (IMRD) structure. Swales noted, however, that Tarone et al., when studying RAs in astrophysics, did not find this IMRD structure. In an update, Tarone et al. (1998: 115) refer to Swales’s hourglass rhetorical structure as an Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 423 adequate format for describing experimental studies; however, they emphasize that it does not apply to RAs in scientific fields where ‘the subject matter does not lend itself to experimentation’. Although they are studying voice in astrophysics papers, Tarone et al. (1998: 115) noted that in logic-driven fields, the rhetorical shape of an RA features logical arguments, which ‘cite observations and draw conclusions’. In this article, we report significant variation from the IMRD structure in mathematics RAs compared to other disciplines. Introduction and Results sections were always pre- sent (in abundance), but Method and Discussion sections were brief or absent. Conclusions were used more in applied than in pure mathematics RAs. Ethnographic data suggests that the macro-structure of math RAs arises out of four factors: the dominance of formalism in research mathematics, the establishment of pre- sumed research methodology, the powerful presence of logic in the discipline and the complicated nature of the mathematical concepts manipulated. In addition, the Conclusion sections in applied math RAs generally contain promotional discourse, reflecting interdisciplinary demands on this area to produce optimized results of potential use to algorithm-based fields including computer science. This study contributes in four ways: 1) to the best of our knowledge, mathematics has received little study using an ESP approach (a recent study on pure mathematics RAs by McGrath and Kuteeva (2012) discusses stance and engagement); 2) it shows how data drawn from multiple sources can illuminate why disciplines write the way they do; 3), it illustrates why novice writers need both disciplinary epistemic and academic discourse competencies to learn to write well in their fields; and 4) it indicates that genre knowl- edge must include understanding of the knowledge construction process in the discipline and how discourse community values establish the context for the RA. Students who are situated in the context will gain a much stronger understanding of the disciplinary dis- course. This study further suggests that mathematics is a discipline because of the way it argues for new knowledge, an implication that highlights the importance of understand- ing the role of argumentation in mathematics. In the following section we briefly explain the data and our methodology. In the third section we present and discuss our findings. The final section summarizes our conclusions. Data and methodology The 30 RAs used in this study (see Appendix B) were obtained through stratified random sampling from these research journals: Discrete Mathematics (DM), Discrete Applied Mathematics (DAM), Journal of Combinatorial Optimization (JCO), Graphs and Combinatorics (G&C) and SIAM Journal of Discrete Mathematics (SIAM). These jour- nals publish major research findings in discrete mathematics. The criteria for selection were representativeness, reputation and accessibility of the journal. DM and DAM are published by ScienceDirect, JCO and G&C by Springer and SIAM by the Society for Applied and Industrial Mathematics; all are indexed by ISI web of knowledge, recom- mended by disciplinary informants and subscribed to by most university libraries. Additional criteria for selecting these journals include topic coverage and target readers, information obtained from the journal homepages. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 424 Discourse Studies 15(4) Table 1.  Some details of the corpus. Number of RAs Number of pages Number of Publication date Number of countries authors represented 30 345 65 2007–2009 21 The following principles ensured a stratified random sample: authors (one article per author was sampled), journals (equal numbers of articles were chosen from each journal), issues (one article was sampled from a given issue), year of publication (equal numbers of articles were chosen from a three-year period, 2007–2009). Thus, author, journal, issue and year of publication were the strata used in sampling the data. To ensure a random sample, article topics were selected that span the branches of discrete mathematics (the study of objects that can assume distinct values and are represented by integers; Renze and Weisstein, 2012). Since the study did not com- pare pure versus applied math papers, we did not use article type as a strata; how- ever, having sampled the corpus, we identified nine applied math papers (DAM2, 3, 4, DM5, D&C5, JCO2, 5, 6, and SIAM2) out of 30. Table 1 summarizes details of the corpus. In our methodology, we adopted a triangulated approach (Candlin and Hyland, 1999) that integrated textual data (description), ethnographic accounts obtained through interviews and discussions with disciplinary specialists (interpretation), and investigation of the structural and social grounds of the writing practices in the discipline as social institution (exploration). To obtain textual data, we used lexico- grammatical signals and content information to characterize the rhetorical struc- ture. When there were few textual signposts to the rhetorical structure or the contents were too technical to identify communicative purpose, we consulted insider inform- ants. Our insider informants were selected based on their high academic qualifica- tions and numerous research publications. These informants also provided ethnographic information by explaining the problems addressed in each paper as well as the introductory material required for each problem, checking samples of the results, and validating our conclusions (that is, we showed informants the coded passages so that they could agree or disagree. In the event of disagreement, we assessed the reasons for their labels and considered their viewpoints in our final judgments). The informants received a short introduction to the IMRD framework and Swales’s ESP approach to genre analysis before checking samples of our rhetorical analysis. Results and discussion As noted, mathematics RAs do not follow the organizational pattern of the IMRD framework (Introduction, Method, Results, Discussion). In fact, they depart consid- erably from it (see Figure 1). Immediately obvious is the absence of both Method and Discussion sections, suggesting that this framework is not a valid model for math RAs. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 425 Figure 1.  Main sections in math RAs. Table 2.  Frequencies of section headings in the corpus. Introduction Complementary Method Results Results Discussion Conclusion Conclusion (GSH) introductions (GSH) (CSH) (GCH) (CSH) (CSH) RAs 30 13 0 2 81 0 3 6 Note: GSH = Generic Section Heading; CSH = Content Section Heading. Another notable difference is the use of section headings (see Table 2). Math RAs routinely use content (rather than generic) section headings. Introduction sections With one exception, all RAs in the corpus used the generic heading of ‘Introduction’ (DAM5 uses ‘Background’ instead). In addition, several articles used compound section headings (e.g. ‘Terminology and introduction’ (DM4) and ‘Introduction and problem description’ (JCO2)). These headings identify content and rhetorical function. Two of the journals, DM and DAM, require authors to include an Introduction section; the others (JCO, G&C and SIAM) do not. The prevalence of Introduction sections in our corpus suggests that writers of mathe- matics RAs (similar to writers in other disciplines) begin by creating a rhetorical space for their research, space that not only highlights the importance of the research and introduces the new results, but also contributes to the argument. In mathematics, persuasion only results when readers and writers share understanding about the concepts being discussed; thus, all introductory sections include clauses that define mathematical concepts and intro- duce symbols. The act of definition is rhetorical here because it simultaneously identifies topic details, clarifies the gap in knowledge being addressed, and creates shared knowledge between writer and reader from which the results arise. In the following examples, the writers use the first paragraph to define concepts and designate symbols: Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 426 Discourse Studies 15(4) Let G = (V,E) be a graph with vertex set V and edge set E. A total dominating set, denoted by TDS, of G with no isolated vertex is a set S of vertices of G such that every vertex is adjacent to a vertex in S. (DM2) In this paper, G is a simple connected graph with vertex set V(G) and edge set E(G) (briefly V and E). For every vertex v∈V, the open neighborhood N(v) is the set N(S)= ∪v∈SN(v) and the closed neighborhood is the set N[v]=N(v). (DAM1) According to Morgan (2005) and Jamison (2000), definitions are essential constitu- ents of doing mathematics. Our informants also noted that the act of definition is a typi- cal communicative purpose for Introduction sections in math RAs in the sense that new conclusions (Move 3, Step 2, Swales’s (1990) CARS model) follow and are deduced from preceding definitions. Definitions, thus, contribute to the communicative purpose of creating a space for research in that they serve as scaffolding upon which new findings are built. We emphasize that the hypothetical nature of mathematics drives this purpose: the objects of research are abstract concepts that are instantiated through precise definition. Thus, a preliminary step involves mathematicians asking readers to suppose the exist- ence of mathematical objects with specific features. The objects are then transformed into symbols that, among other things, allow mathematical manipulation. For example, ‘G’ above is defined as ‘a graph with vertex set V and edge set E’; the 10-word definition is condensed into the single letter, ‘G’. G is then used in the formulas and equations that follow. Although introduced throughout an RA, mathematical concepts and symbols are criti- cal in Introductions where authors must establish agreement with readers on the nature of mathematical concepts and properties. Once agreement is reached, authors can proceed to making and proving their knowledge claims (Foss et al., 2002). Complementary introductions Over one-third of the Introductions (36.66%) include additional sections located before results and labeled with content section headings (13 additional sections), for example, ‘Definitions’, ‘Known results and more definitions’ and ‘Construction of prism fixers’ (three sections in DM3) and ‘Man-Exchange Stable Marriage’, ‘The Gale-Shapley algo- rithm’ (two in JCO2). Articles from each journal included additional sections, indicating it is a conventional structure. In these sections authors define specialist terminology, introduce notations, describe the problem and review previous research. For example, the section two heading in DM2 states its purpose: ‘Known results on total domination’ (i.e. reviewing previous results). It includes seven theorems and citations followed by authors’ comments on those results. The section reviews previously published results, results that two disciplinary informants noted were substantial to ‘total domination in graphs’, the paper topic. The review con- textualizes the authors’ current results on total domination, a communicative purpose identified as Move 1, Step 3 in Swales’s (1990) CARS model for RA Introductions. Then the authors incorporate this previous work into the solution that they present for the Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 427 problem addressed in DM2. When authors in math need to review a substantial body of literature, they present it as a separate section (not part of the general Introduction), and if this previous research is used in solving the current problem, they bold the heading to emphasize the section. As noted, DM3 presents three intervening sections between Introduction and Results. As the section headings, textual signals and the informants’ data indicate, these sections are prompted by objects under study that require successive definitions and substantial literature review. The Complementary introductions allow writers to organize the defini- tions and symbols while maintaining the clarity of the Introduction. Writers also ensure the results are easily located by presenting them as freestanding sections. Descriptive headings not only allow writers to highlight the research objects’ features; they also enable readers to more easily conceptualize the objects. These sections help establish the context for the mathematical argument and prepare readers for the results that follow, a function that other researchers, including Lim (2011), have established. All 13 Complementary introduction sections shared this pattern. General definitions and reviews of well-known results generally appear in the Introduction; however, when the mathematical objects or previous research are foundational to the paper, they appear as separate sections (i.e. Complementary introductions). Absent sections In our corpus, Method and Discussion sections are absent (see Figure 1). This departure from the IMRD framework indicates that, not surprisingly, mathematics lacks an empiri- cal basis. At the same time, exploring why Method and Discussion sections are absent in math RAs helps to articulate a fundamental understanding of mathematics as a disci- pline. We considered the communicative purposes reported for these sections in genre studies, but little work has been published to date on Method sections. Swales (2004) suggests significant demarcations between disciplines in terms of their research method- ologies, noting that the typical structure of Method sections includes a description of the materials used, the procedures adopted, the apparatus employed and the statistical analy- ses chosen. He further posits a relationship between discipline and length of Method sections: methods are ‘clipped’ in many hard sciences; elaborated in education, psychol- ogy and the social sciences; and of intermediate length in language sciences, public health fields and earth sciences. The absence of need to explain methodology in mathematics is interesting because it differentiates this field from both hard and soft sciences. In hard sciences, Method sec- tions describe the physical actions performed by researchers prior to and during scien- tific experiments. These descriptions enable readers to validate (and possibly replicate) the results reported. In contrast, research activities in math are mainly cognitive, a criti- cal difference between math and experimentally oriented fields. The absence of a Method section also implies agreement among members of the discourse community on how to solve mathematical problems; that is, to a great extent insiders presume the method. This point supports Brett’s (1994) contention that the extended Method sections in sociology RAs (compared to shorter Method in the hard sciences) indicate less agreement on the methodological practice in the discipline. Its absence in math RAs also points towards Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 428 Discourse Studies 15(4) the extended history of mathematical problem solution, a history with well-established and taken-for-granted methods of logical deduction and induction. At the same time, absence of a Method section does not mean methods are absent in mathematics RAs. In fact, authors allude to methodology in their proofs. The following example explicitly identifies the authors’ method: In our proof we will use the method of cyclic permutations developed by Gyula O.H. Katona in [7]. (Bold in original, underline added, G&C2: 359) The first sentence of the proof identifies the method but refers readers to a citation for more information. Other examples suggest that mathematicians signal method briefly in their texts: We prove this by induction on the order n of the tree. (Underline added, DAM1, Proof of Theorem 17) This statement indicates the proof technique called ‘mathematical induction’. Numerous similar instances are discernible in our corpus. Another common method is proof by contradiction. Again, this technique is linguisti- cally signaled: Proof by contradiction. (Bold in original, underline added, DM1, Proof of Lemma 3.6.) We show by contradiction that . . . . (Underline added, JCO5, Proof of Lemma 2) In addition to lexemes referring to method, some discourse markers signal procedural descriptions of the problem–solution process. For example, ‘first’ and ‘then’ are linguis- tic markers that guide readers through the procedure of proof. The following example uses ‘first’ and ‘then’ to describe a chronological methodology: We will first prove that . . . . We will then prove that . . . . (Underlines added, DAM4, Proof of Lemma 3) By signaling the procedure, mathematicians highlight the knowledge and agreement about methodology shared among disciplinary insiders. Since method is presumed in mathematics, it is reduced to a phrase or procedural account embedded in the proof. In other words, oblique references to method are judged adequate by disciplinary insiders. However, different mathematical results within an article can require different tech- niques of proof and procedures. That is, each proof may be developed using a unique procedure. However, our data also suggest procedural similarities exist among proofs within a paper. In such cases the authors achieve conciseness by referring to proofs presented earlier, as in the following example: To complete the colouring we note that . . . and so the same argument applies. (Underline added, DAM5, Proof of Theorem 3) Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 429 The question arises of whether it is possible to summarize these independent descriptions within a distinct section, and if so, whether it would compromise the readability of the paper, creating ambiguity in the proofs. Our subject specialists suggest not. They note that because the basic methodology is widely assumed in mathematics, brief linguistic cues activate shared knowledge and keep readers on track. As long as mathematicians base their claims on logic and argue inductively or deductively through a piece of proof, readers do not want more detailed information. In other disciplines, the method is used to validate results, but in mathematics, results are valid as long as they are logical deduc- tions from premises. Therefore, the absence of Method sections in math RAs arises from a community-established norm that presumes readers know standard disciplinary research methodology. Not only are Method sections absent, but Discussion sections are also absent in our corpus. That is, while no separate Discussion section is present, a discussion of results does take place. Discussion-like activities appear within argumentative moves in Results sections: accounting for, commenting on, extending or delimiting, evaluating or exem- plifying findings. These communicative purposes are typical with ‘Discussion’ sections in other disciplines; however, in mathematics RAs the discussion is brief and integrated into the presentation of results. The following passage, very much resembling discus- sion, succeeds the proof of Theorem 3 in DAM2: In view of the previous result we can say that the diameter . . . is at most 3. Therefore T(B) must be one of the following: . . . (Underlines added, DAM2: 2743) The introductory phrase signals that the authors are elaborating the preceding result. The adverb ‘Therefore’ signals that they are drawing conclusions about the first statement. In this passage, the principal communicative purposes are accounting for findings and deducing further results. In the next example, the writers also evaluate the result that they presented in Corollary 3: Therefore, we conclude that the lower bound of Corollary 3 is better than the information theoretic bound. (Underlines added, DM5: 5936) The lexemes ‘Therefore’ and ‘conclude’ and the adjective ‘better’ suggest that the sen- tence evaluates the knowledge claim established in the corollary. While acts of evalua- tion typically appear in the Discussion sections of RAs in other disciplines, in mathematics the significance of results is noted immediately. If we consider the communicative func- tion of Discussion sections in general, we can see why discussion is presented with results in mathematics. In other disciplines, results may be understood differently by other researchers; therefore writers in those disciplines explain their understanding of results – they argue for their interpretation, and this argument appears in the discussion section. In contrast, results in mathematics require little interpretation. The proof of a mathematical hypothesis is demonstrative in itself. Anyone who looks at the proof either accepts or rejects it. If they reject it, it is because of a flaw in the reasoning (that is, the proof is executed incorrectly). If the proofs are based on the right premises and logically argued, the results are accepted; hence they are persuasive (i.e. convincing) in Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 430 Discourse Studies 15(4) themselves and require little, if any, discussion to convince readers. That is, the proof itself persuades readers, not the discussion of the proof: if the proof is not logical, it will not be judged persuasive regardless of the discussion presenting it. An informant con- firmed that this applies to all proofs, regardless of their weight. Foss et al. (2002) note that most people are inclined to accept claims based on logic. Mathematical findings become persuasive because they are products of logical reasoning. They need little interpretation (or discussion) because they are demonstrative by themselves. Results section Results sections comprise nearly two-thirds of the total number of sections in the corpus (61.5% of a total of 135 sections) (see Figure 1). Results sections were designated based on those labeled as such. Two characteristics were notable: 1) content section headings dominated the presentation of results (see Table 2), and 2) multiple results sections were presented (see Figure 2). Only two of 83 Results sections were generically labeled ‘Main result’ (DM2 and SIAM5); the rest used content headings. DM2 contained three Results sections in addition to the first, labeled ‘Main result’. The Main result section begins ‘We shall prove’, and presents a theorem and corollary but no proof: in fact, this section merely announces the main result. Further analysis reveals that the actual proof for the theorem appears two sections later, labeled ‘Proof of Theorem 8’. One section, called ‘Cost function’, separates the ‘Main result’ and its proof. It begins thus: Before presenting a proof of Theorem 8 we introduce the concept of a cost function … (DM2: 3494) Clearly, the authors believe that readers must understand the ‘concept of a cost function’ to follow the proof. Our specialist informants identified this concept as a new contribu- tion by the authors: since it is in itself a result, it is highlighted with its own heading. Figure 2.  Number of RAs with multiple Results sections. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 431 Setting it off emphasizes its importance and draws it to reader attention. These authors use this sectioning strategy rhetorically to highlight the novelty of the concept and the creativity of this work. It also improves readability. The second case that uses ‘Main result’, SIAM5, employs its second and third Results sections to present solutions to previously published results that have not been solved. That is, the ‘Main result’ section presents the authors’ main solution, but two follow-up Results sections announce solutions for problems (i.e. ‘fill a gap’, to use Swales’s terminology (1990, 2004)) that were unsolved hitherto. Disciplinary informants explained that these solutions are new contributions to graph theory research, and it is conventional to present them in independent sections. In addition, the sectioning strategy emphasizes the new results. As illustrated, math RAs typically include multiple results sections; in our corpus, each RA included 2.8 Results sections (on average). With the exception of JCO5 and DM3, all of the papers included more than one section of results (while DM3 presented a single Results section, JCO5 presents the results as four sub-sections under one heading). Figure 2 shows that 22 out of 30 RAs in our corpus (73.33%) use two or three Results sections, indicating that presenting results in two or three sections is typical in math RAs. Subdividing results allows mathematicians to highlight their contributions and aid reader comprehension. As mentioned, the act of defining a novel function is considered a new research find- ing in mathematics. In such cases, authors devote an independent section to that defini- tion. Researchers also use multiple Results sections when they have been able to solve a problem for different quantities. For example, the authors in DM5 present lower and upper bounds for the graphs, as well as solving the problem for some complete graphs; in each case they present that work in its own section. They orient readers to the results’ organizational structure: In Section 3 we start with the proof of upper bounds for . . . Section 4 is devoted to the proof of the sharp upper bound for . . . In Section 5 we consider the complete graph Kn. We provide a sharp lower bound for . . . and improve the upper bound of Section 4 for . . . (DM5) The section headings in DM5 declare the rhetorical arrangement. Each heading, ‘3. Lower bounds’, ‘4. Upper bounds’, ‘5. The complete graph Kn’, indicates an autono- mous result. The author of SIAM5, an applied math paper, uses a similar rhetorical structure for his two Results sections. In the first one, ‘Characterizing theorem’, he characterizes and solves the problem that he regards as the underlying theorem for the algorithm he has developed. He also announces this result in the Introduction section and then announces a further result – the algorithm that arises from the theorem: Employing this characterization, we present a[n] … algorithm for … (SIAM5: 637) The algorithm appears in the second Result section, labeled ‘3. Monte Carlo algorithm’. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 432 Discourse Studies 15(4) According to disciplinary insiders, obtaining an algorithm for solving a real world problem in other disciplines like computer science, electronics or biology is a character- izing feature of applied mathematics. The algorithm is regarded as the key finding, and as such it normally appears in an independent section. Mathematicians who deal with complex or multifaceted mathematical objects often subdivide their results to communicate more effectively the features of the topic. They probably have additional motives for locating important results in individual Results sec- tions; study of a broader corpus would undoubtedly reveal these. Nonetheless, these findings indicate that authors section their results strategically and use section headings to highlight aspects of their work and to aid reader comprehension. Conclusion section Only nine RAs in the corpus include a Conclusion section (see Figure 1). These sections use variations on generic headings: ‘Conclusion’ is used three times (in DM2, DAM4, JCO5). Other headings include ‘Concluding remarks’ (in DAM3, G&C5), ‘Summary and Conclusion’ (in JCO2), ‘Other remarks and further work’ (in SIAM2), ‘Conclusions and future work’ (in SIAM6) and ‘Comments and Open Problem’ (in G&C2). Six of these conclusions appear in applied mathematics RAs (DAM2, 3, 4, JCO2, 5 and SIAM2). Disciplinary informants noted that Conclusion sections are more common in applied math papers and rare in pure math RAs. The rhetorical purpose of these sections ranged from reviewing the results, emphasiz- ing the significance of the problem or an application of the results, evaluating and/or interpreting results, to suggesting further research on the subject. These purposes resem- ble those reported for Discussion and Conclusion sections in other disciplines. The following examples illustrate some of the strategies used in the Conclusion sections. Reviewing the results: In this paper we considered … graphs. We proved that… (Underlines added, DAM2: 2752) Emphasizing the significance of the problem or applications: In this paper we have introduced . . . motivated by a significant practical application. . . . In the context of . . ., this . . . problem could still be significant. (Underlines added, JCO2: 358) The concept of a balanced decomposition number f .G/ seems to have many other applications. (Underlines added, DAM3: 3344) Evaluating results: This result was quite surprising since disjoint path problems are notoriously hard in directed graphs. (Underline added, DAM4: 96) Interpreting results: We may be able to regard the concept of … as an extension of . . . . (DAM3: 3344) Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 Graves et al. 433 Suggesting further research: One interesting problem is to extend our results to . . . of these graph families . . . Finally, it would be an interesting problem to characterize . . . . (Underlines added, G&C5: 166) Except for ‘reviewing the results’, these communicative purposes are specific to applied math RAs, which generally serve research in engineering, health science, biology and other scientific disciplines. That is, readers from these disciplines fol- low research in applied mathematics to identify algorithms and models useful to their own work. Applied mathematicians use a Conclusion to highlight the optimal- ity of their algorithms to entice researchers from other disciplines to apply their results in subsequent work. Although disciplinary informants confirm this interpreta- tion, our corpus size is too small to make extensive claims about using Conclusions in applied math. Conclusion In this article we analyzed the organization and rhetorical structure of sectioning in math RAs. This analysis shows that IMRD, the dominant model in empirically oriented disci- plines, is (unsurprisingly) not the framework for math RAs. This confirms Tarone et al.’s (1998) conjecture that Swales’s hourglass model may not apply to mathematics RAs. While researchers in empirical sciences must describe their procedure and explain the rationales for using a particular method (Bazerman, 1988; Swales, 1990), mathematicians do cognitive work that relies on logical rules of deduction and induction. Perhaps also unsurprisingly, the structure of RAs in this discipline reflects this ontological tradition. In our analysis, we have tried to explain why mathematicians organize and present their knowledge claims in particular ways. As the community of a logic-driven disci- pline, mathematicians use an Introduction-Results macro-structure in their RAs. The Introduction not only creates space for new work and emphasizes its importance, it also defines the mathematical concepts that constitute the subjects of research. In fact, pre- senting definitions as the necessary groundwork for arguing knowledge claims is a long- established tradition in mathematics. This same tradition of relying on logic also accounts for the absence of a Method section. Again, reliance on logic also obviates the need for discussion, hence its absence from the RA framework. Instead writers in mathematics explain their findings through brief statements integrated directly into the Results sec- tions. When mathematicians wish to highlight the applicability of their work to scholars in other disciplines, they locate this appeal in a brief Conclusion section. This analysis highlights the link between RA structure in mathematics and its discipli- nary traditions; it further shows that the RA is a historically based genre. It also suggests that when teaching academic writing in disciplines, we must include explanatory accounts of why disciplines write the way they do (Bhatia, 2004). As Hyland (2000: xiii) explains, academic writing is more than proclaiming research; it is ‘evidencing a sophis- ticated awareness of how disciplinary cultures textualize that research into knowledge’. This awareness, we believe, can only help novice writers write more successfully in the discipline. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 434 Discourse Studies 15(4) This study suggests that teachers and students of writing must pay much more atten- tion to the structural and rhetorical organization of RAs in each discipline. Students require structural information to help them identify and understand the key markers that constitute an effectively organized argument and structural framework for RAs in their disciplines. This study also shows that macro-rhetorical information is important in learning to write in a particular discipline because it enables writers to connect traditions in the discipline with choices about structure in its genres. Acknowledgements The authors would like to acknowledge the financial support of both the International Students Centre (ISC) at the University of Malaya and the Writing across the Curriculum Program at the University of Alberta. The authors also appreciate the generous contributions of time and expertise from the mathematicians who acted as disciplinary informants for the work presented here. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. 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Author(s) RA section studied Disciplines Language(s) Basturkmen (2012) Discussion Dentistry English Del Saz-Rubio (2011) Introduction Agricultural sciences English Kanoksilapatham (2011) Introduction Civil engineering English Parkinson (2011) Discussion Physics English Soler (2011) Titles Biology and social English and Spanish science Lim (2010) Discussion Applied linguistics and English education Hirano (2009) Introduction Applied linguistics Brazilian Portuguese and English Basturkmen (2009) Discussion Language English Bruce (2009) Results Sociology and organic English chemistry Bruce (2008) Method Physical sciences and English social sciences Pho (2008) Abstract Applied linguistics and English educational technology Ozturk (2007) Introduction Language English Van Bonn and Swales (2007) Abstract Language English and French Monreal et al. (2006) Section headings Computer sciences English Lim (2006) Method Human resource English management Kanoksilapatham (2005) All sections Biochemistry English Samraj (2005) Abstract and Biology English Introduction Appendix B Articles in the corpus DM1. Hoffmann-Ostenhof A (2007) A counterexample to the bipartizing matching conjecture. Discrete Mathematics 307(22): 2723–2733. DM2. Favaron O and Henning MA (2008) Bounds on total domination in claw-free cubic graphs. Discrete Mathematics 308(16): 3491–3507. DM3. Gibson RG (2008) Bipartite graphs are not universal fixers. Discrete Mathematics 308(24): 5937–5943. DM4. Meierling D and Volkmann L (2009) On the number of cycles in local tournaments. Discrete Mathematics 309(8): 2042–2052. DM5. Gerzen T (2009) Edge search in graphs with restricted test sets. Discrete Mathematics 309(20): 5932–5942. DM6. Bielak H (2007) Chromatic properties of Hamiltonian graphs. Discrete Mathematics 307(11–12): 1245–1254. 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Graphs and Combinatorics 23(2): 165–182. G&C2. Bernáth A and Gerbner D (2007) Chain intersecting families. Graphs and Combinatorics 23(4): 353–366. G&C3. Chandran LS, Kostochka A and Raju JK (2008) Hadwiger number and the Cartesian product of graphs. Graphs and Combinatorics 24(4): 291–301. G&C4. Ghebleh M, Goddyn LA, Mahmoodian ES and Verdian-Rizi M (2008) Silver cubes. Graphs and Combinatorics 24(5): 429–442. G&C5. Bonomo F, Durán G, Maffray F, et al. (2009) On the b-coloring of cographs and P4-sparse graphs. Graphs and Combinatorics 25(2): 153–167. G&C6. Cui Q, Haxell P and Ma W (2009) Packing and covering triangles in planar graphs. Graphs and Combinatorics 25(6): 817–824. JCO1. Cardoso D, Kamiński M and Lozin V (2007) Maximum k-regular induced subgraphs. Journal of Combinatorial Optimization 14(4): 455–463. JCO2. Irving R (2008) Stable matching problems with exchange restrictions. Journal of Combinatorial Optimization 16(4): 344–360. JCO3. Goddard W and Henning M (2007) Restricted domination parameters in graphs. Journal of Combinatorial Optimization 13(4): 353–363. JCO4. Vanetik N (2009) Path packing and a related optimization problem. Journal of Combinatorial Optimization 17(2): 192–205. JCO5. Liu C, Song Y and Burge L (2008) Parameterized lower bound and inapproximability of polylogarithmic string barcoding. Journal of Combinatorial Optimization 16(1): 39–49. JCO6. Messinger M and Nowakowski R (2009) The robot cleans up. Journal of Combinatorial Optimization 18(4): 350–361. SIAM1. Berman KA (2007) Locating servers for reliability and affine embeddings. SIAM Journal on Discrete Mathematics 21(3): 637–646. SIAM2. Takata K (2007) A worst-case analysis of the sequential method to list the minimal hitting sets of a hypergraph. SIAM Journal on Discrete Mathematics 21(4): 936–946. SIAM3. Dankelmann P, Simon M and Henda CS (2008) Average distance and edge-connectivity I. SIAM Journal on Discrete Mathematics 22(1): 92–101. SIAM4. Korner J, Claudia M and Gabor S (2008) Graph-different permutations. SIAM Journal on Discrete Mathematics 22(2): 489–499. SIAM5. Krotov DS and Vladimir NP (2009) n-Ary quasigroups of order 4. SIAM Journal on Discrete Mathematics 23(2): 561–570. SIAM6. McClosky B and Illya VH (2009) The Co-2-plex polytope and integral systems. SIAM Journal on Discrete Mathematics 23(3): 1135–1148. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014 438 Discourse Studies 15(4) Author biographies Heather Graves is an Associate Professor of English and Film Studies at the University of Alberta, in Edmonton, Alberta, Canada. In 2011 she was the first Scholar in Residence for Arts Research in Nanotechnology at the Canadian National Institute for Nanotechnology. Her research interests include the rhetoric of science and argument in academic discourse. Shahin Moghaddasi is a PhD candidate in the Faculty of Linguistics, University of Malaya. She is currently a visiting graduate fellow in the Department of English, University of Alberta. Her research interests include genre analysis, academic writing and discourse studies. Azirah Hashim is a Professor at the English Department, Faculty of Languages and Linguistics, University of Malaya and currently the dean of the Humanities and Ethics Research Cluster. Her research interests are in Academic and Professional Communication, Language and Law, and Language Contact. Downloaded from dis.sagepub.com at Universiti Malaya (S141/J/2004) on January 12, 2014