Practice and conceptions: communicating mathematics in the workplace
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Abstract
The study examined the experience of communication in the workplace for mathematics graduates with a view to enriching university curriculum. I broaden the work of Burton and Morgan (2000), who investigated the discourse practices of academic mathematicians to examine the discourse used by new mathematics graduates in industry and their perceptions of how they acquired these skills. I describe the different levels of perception of discourse needs and of how they gained the necessary skills. At the lowest level, they learnt through trying out different approaches. At the next level, they were assisted by colleagues or outside situations. At the highest level, a small group viewed communication and interpersonal skills as a scientific process and stood back and used their “mathematical” observation skills to model their behaviour. These graduates did not appear to have systematically studied communication as part of their degree and they were unaware of the power of language choices in the workplace. Those who were working as mathematicians had to come to grips with explaining mathematical concepts to a wide range of people with varying mathematical skills but who generally were considerably less skilled in mathematics. The study revealed that these graduates were seriously underprepared in many aspects for joining the workforce. Many found it difficult to adapt to dealing with colleagues and managers, and developing communication skills was often a matter of trial and error.
Keywords
Graduates Mathematical communication Phenomenography Professional preparation Workplacerelated mathematics1 Introduction
We interpret reality through language, according to the discourse analysts. In my case, I interpret the world through the lens of mathematical discourse. For most of my academic career, I have been fortunate enough to be engaged as a mathematician, communicating mathematics to other mathematicians and to students who have been exposed to the discourse of mathematics through their schooling. There are those, however, who qualify as mathematicians from university without any specific training in communication skills, but then find themselves going out into the world and needing to communicate mathematical concepts with people who have not been as widely exposed to mathematical discourse. How do they fare? Very little attention has been focused on the reality of functioning within the workplace from a learning and teaching perspective. Research on employment from the graduates’ perspective is another neglected area (Johnston, 2003).
Most of the reported research about mathematics and language comes from mathematicians (mostly academics) writing within the discipline of mathematics, that is, they are writing about communicating mathematics within an area of discourse where the rules have already been established. This scenario essentially presumes that communication is with other mathematicians. The teaching and learning of tertiary mathematics is set within a similar framework, where although the mathematics may be communicated across disciplines—such as to engineering students—a commonality of both mathematical understanding and mathematical language is assumed on the basis of prior studies. However, in the nonacademic workplace mathematics graduates are generally communicating mathematical concepts to nonmathematicians; even where the recipients may have received mathematical training, it is at a considerably lower level than that achieved by mathematics graduates.
Discussions about mathematical discourse are also founded on a conscious assessment of language. In general, however, mathematical language is absorbed by secondary and tertiary students by doing mathematics, since it is rare to find instances where training in communications has been an intrinsic part of mathematical learning. It is presupposed that by learning the mathematics, students will also learn the discourse; and then will automatically be able to communicate mathematical ideas to a wider public.
From the point of view of an organisation, an employee who can communicate well with others makes a more significant contribution to that organisation. Employers see communication skills as the most important graduate capability (Hoyles, Wolf, Kent, & MolyneuxHodgson, 2002). From the perspective of the individual, the power of language choices can help you get a job, adapt to the job and progress in a career (Burton, 2004).
I set out to investigate the forms of discourse that mathematics graduates find themselves using in the workplace and what indeed are their conceptions of mathematical discourse. And since my own context is that of an educator, I address the question: do graduates recall explicit learning or teaching of the mathematical communication they need for their employment?
2 Language and mathematics
2.1 Mathematical discourse
In this paper, I use the phrase mathematical discourse to refer to the uses of language in university mathematics learning and teaching and in professional life. Discourse is a broader concept than language because it also involves all the activities and practices that make up mathematics as a profession, such as through textbooks, computer packages, and through application of mathematical knowledge. Mathematical discourse is distinct from other types of discourse simply because mathematics is quite distinct from other subjects or disciplines and the things that mathematicians do are distinct from what other academics and professionals do: each discipline requires specific elements and styles of communication.
Naturally, there are variations of the form depending on the mathematical and communication skills of the presenter and the receiver, but certain features distinguish mathematics discourse from other forms. For instance, it is often highly symbolic and incorporates graphic representations as well, as summed up by Hammill (2010) who looked at research on interactions between text and symbols. She also addresses the role of graphic representation and notes that: “Symbolic expressions need to be created to model real world problems and then graphs constructed from the symbolic models” (p. 1). Mathematical discourse differs from natural language in other ways such as the following: it is more economical, it encapsulates concepts in a way that clarifies connections, and it is easier to manipulate mathematical objects using their own syntax.
Mathematical symbolic language is not universal and unchanging, that is, it has been invented and developed by mathematicians to do a particular job. It could be argued that the converse is also true: that the development of the symbolic language has influenced mathematical thinking. The relationship between discourse and conceptualisation is a twoway street with each relying on the other. O’Halloran (2005) is one of the recent theorists who have been pursuing the connections between symbolism and visual images (or graphic representations) and thinking in the development of mathematical ideas, and who leads us into the everexpanding territory of computer representation.
It is important to note, nonetheless, that the use of symbolic language has its limitations. It is easier to follow if it is expressed in ways that follow the conventions of natural language and, moreover, it is usually framed within explanatory text. A mathematician uses natural language to help other people—whether mathematicians or nonmathematicians—to understand their mathematical thinking. What we find is that mathematicians mix symbolic language with English (or another language) in different proportions and in different ways depending on what they think their audience will be comfortable with. Thus, an expert in analysis, for example, may use a greater proportion of symbolic language to communicate with a similar kind of expert than he or she would with a mathematician whose field is statistics. The real skill comes in attempting to convey mathematical concepts in predominantly natural language without losing the integrity of the mathematics.
2.2 Examining mathematical discourse
On the whole, students have worked only on relatively short, routine problems for which little elaboration or explanation is required…In contrast, “real” mathematicians tend to work on relatively substantial and often original problems. Their anticipated audiences are expected to be genuinely interested in knowing the results and to need to be persuaded of the correctness of the results. (Morgan, 1998, p. 2)
The point is made that it is the representation of the mathematics to others that is problematic.
A group of French mathematics educators, exemplified by Duval (1999, 2001), have extended this work. Duval argues that we can gain access to mathematical objects only through their representations. This idea has led to a research school that investigates the ways that students develop their mathematical understandings with different representations. DurandGuerrier (2004) sheds light on the fact that details in mathematical discourse can lead to deep misunderstanding in students; how much more problematic IS IT then for explanations to nonmathematicians.
Gestures can perform a role here in facetoface mathematical discourse, and this has been receiving some recent attention; for instance, see the special issue of Educational Studies in Mathematics on this subject (“Gestures and Multimodality in the Construction of Mathematical Meaning”, 2009, vol. 70, no. 2). Thus mathematicians—in common with any other human engaging in facetoface discourse—will use gesture to enhance the meaning of what we are trying to convey. Sfard (2009) suggests in this special issue that the wave of interest in gestures across many disciplines has arisen in part because of developments in technology (such as modes of recording and analysing), but she also recommends caution as it is essential “to make a clear distinction between the roles of speech and that of gesturing in constituting and sustaining mathematical discourse” (p. 192).
At the professional level, Burton and Morgan (2000) examined the ways that academic mathematicians write. They argue that writing plays a critical role in many mathematical practices and that guidance in the formation of these texts is essential: “knowledge of the forms of language that are highly valued within mathematical discourses and the effects that may be achieved by various linguistic choices…would empower them,” to make choices, to break conventions and express their own personality (p. 431).
Mathematics is created by communicating, that is, mathematics is created in the act of communication…mathematics is both enabled and restricted by the conventions of communication. (p. 173)
Learning mathematics, and doing mathematics, involves talking mathematics: the more we talk mathematics, the better we will learn it and do it. (p. 174)
If we support the premises of these authors that mathematics is developed and learnt through communication and talking, why is the explicit learning and teaching of mathematical communication skills so thin in mathematics curricula? How does a mathematician trained in the use of symbols then explain the mathematics to someone who has not been so trained? This question has received little attention. Beyond the arena of the academic mathematician, the main focus of research has been into the use of mathematics in the workplace (or the need for greater application of mathematical skills) and employers’ expectations of graduates (such as Selden & Selden, 2001; Hoyles et al., 2002). Kent, Hoyles, Noss and Guile (2004) have been doing a study of the needs of industry and they found an increase in the requirement for mathematical literacy amongst employees and for skills of mathematical communication in the workforce. However, this is a rare mention of communication skills and it was raised in a minor way regarding potential problems: “The manager’s model (assuming he/she has a technical/engineering background) will tend to be based on formal mathematical models; the operator’s model is far more based on the physical experience of production …” (p. 8).
2.3 Language and curriculum
A major influence on the development of language and discourse initiatives in curriculum design has been the work in systemic functional analysis theory and practice (for instance Halliday (1985, 1991)). There are two main aims fundamental to integrating language and disciplinary studies that have been used to develop curriculum. The first is exemplified by such articles as Building Bridges: Using Science as a Tool to Teach Reading and Writing (Nixon & Akerson, 2002), where science and mathematics are used as a vehicle to teach language. The second is where language is used to teach the mathematics/science, for example Reading and Writing to Learn Science (Glynn & Muth, 1994). These articles are at primary and secondary school levels respectively.
…lack of explicit attention to form and its crucial function in shaping successful communication leaves students to wallow in their own limitations, unable to partake of the cultural traditions into which they are (hopefully) being inculcated through their schooling. (p. 161)
Morgan (1998) suggests there has been insufficient attention paid to the divergence between the experience of language by students and professional mathematicians within curriculum design at university, with the major focus being on difficulties children may have with mathematical language (p. 3). In a similar vein, DurandGuerrier (2004) argues that the detailed use of language has not been seriously addressed in mathematics education and in Sweden, Österholm (2005) has concluded that there is a “need for explicit teaching of reading symbolic texts, that is, more direct practice is needed for the development of the more contentspecific literacy skills” (p. 342).
Despite the research into discourse, little is known about how mathematics graduates integrate their knowledge of mathematical discourse into a nonacademic workplace and how they learn to communicate mathematical concepts to colleagues and clients.

How do mathematics graduates use discourse to communicate mathematical ideas with a range of audiences?

What is their intention (implicit or explicit) in the communication?

How do graduates learn to communicate mathematical ideas?
3 Research design
3.1 The methodology
Description of participants
ID  Sex  Majors  Job description  Work area 

Angie  F  Maths, finance  Loan advisor  Banking 
Boris  M  Pure, applied  Cryptographer  Security research 
Christine  F  Pure, applied  Police officer  Police 
David  M  Maths, finance  Dealer bank treasury  Banking 
Evan  M  Maths, finance  IT  Banking 
Fredrik  M  Maths, physics  Technical officer  Hospital research 
Gavin  M  Applied  Climate modelling  University 
Heloise  F  Stats, OR  Logistics analyst  Industrial 
James  M  Maths, finance  Corporate treasury  Insurance 
Kay  F  Statistics  Statistician, tutor  University 
Leah  F  Statistics  Clerk  Government 
Melanie  F  Applied  Musician  Entertainment 
Nathan  M  Applied, IT  IT development  Selfemployed, industry 
Paul  M  Maths, finance  Trading risk management  Banking 
Roger  M  Pure, applied  Modeller, programmer  Geological surveying 
Sally  F  Statistics  Statistician  Insurance 
Thi  F  Engineering, maths  Loan support  Selfemployed, bank services 
William  M  Actuarial  Instructional designer  University 
Interview questions
1. How would you describe the work that you do? In what ways do you use mathematics in this? 
2. In what ways do you use mathematics to communicate ideas? Examples for prompting if necessary: reading graphs, using formulas in Excel. From others? Produced by you? 
3. Is mathematical communication different to other forms of communication? 
4. How did you learn these communication skills? 
5. In what way has studying mathematics at university level prepared you for work? 
6. What else helped you to make the transition to professional work? 
7. What do you think should be in a university course to help you to make the transition to professional work? 
D: …And plus it’s very transparent. Something like that is very transparent and it’s very readily and easily explainable. Anything that cannot be readily explained and anything that’s not transparent is not well accepted by management.
I: Transparent: define what you mean.
D: Transparent. It’s just like doing a weighted average, that’s very transparent because you can very easily, very, very easily see what it means. You put numbers on paper and very easily show somebody in a very easy example as to how the whole process works. Quite honestly coming up with some of the stuff, I mean some of the stuff is readily available on Excel, […] just descriptive statistics, they’re not well understood by people. People, they just can’t, they can’t visualise it, they need to be able to visualise it to understand it.
I: So transparent is kind of like if you are able to explain something that your manager understands, that’s transparent.
D: That’s right. And he’s an intelligent man, but his background’s not quantitative, but he’s quite comfortable with quantitative ideas. But I don’t think I’d be able to take him to the […] calculus, that’s not his cup of tea.
I: What about graphs?
P: No. In the most recent role at the […] bank, there was a plan that [if] you ever handed across numbers it’s only ever graphs basically because you can’t sell, people, if they are not maths literate, want to see a picture. [However] doing structured finance, never did any graphs. Where I am, very occasionally if I want to pretty up a report or if it’s going to say, Treasury’s level or something, then yeah, but apart from that, no.
I: But you don’t use graphs at all now because people understand the maths?
P: Well people are more interested in the numbers than anything else.
I: O.K. And what about graphs, do you show them graphs?
T: We have the ability to show them graphs because there’s a lot of online material that you can use that is already available on bank sites and any other finance sites. It shows their payments on how it is reduced, and, if you pay this much per month this is how you can reduce your payments and this is the amount of money that you can save by the end of 30 years or 25 years.
I: And how do they find that?
T: How do they find that? They like the graphical representation more than the…just the numerical.
I: Alright. So in what ways do you use your maths or stats or …?
L: Bugger all!
Many participants also provided samples of texts they had used and/or produced in the workplace. These texts as well as the transcriptions of the interviews were examined for a factual analysis of the range of communication skills used by the graduates, which is presented below in Section 4.1. The transcribed interviews were further analysed using the phenomenographic paradigm (Marton, 1994; Marton & Booth, 1997; Akerlind, 2005; Prosser, 2000), based on illuminating the different ways that they used and learnt mathematical discourse skills note that for the purposes of a phenomenographic study, 15–20 interviews are sufficient (Kvale, 1996).
3.2 Phenomenographic method
Distinctly different ways of experiencing the phenomenon discussed in the interview are the units of analysis and not the single individuals. The categories of description corresponding to those differing understandings and the logical relations that can be established between them constitute the main results of a phenomenographic study. (p. 4424)
The outcome of a phenomenographic study is often a hierarchical set of logically related categories, from the narrowest and most limited to the broadest and most inclusive. This is referred to as the outcome space for the research—the categories sit within this space and attain meaning in reference to each other and to the whole set of data.
The starting point is thus the identification of common themes expressed by participants, which are then grouped into categories through iteration. This may be done by manual coding of segments of text, or using an appropriate software programme. I used NVivo® software for the initial coding and then combed through the transcripts several times, reassessing the original themes and critical differences between them.
Many themes emerged about the nature of mathematical work done by these graduates and their difficulty in finding appropriate employment. There were themes around their transition to professional work and the kinds of learning that would enhance their transition. Graduates also reflected on what was missing in their undergraduate education. In this article, I am reporting on the main finding, which was the complexity of mathematical communication with nonmathematicians in the transition to work.
I coded the transcripts based on the themes that emerged from the data, then looked for overarching patterns. I applied this technique for both the conceptions of communication, and how they learnt mathematical discourse. This revealed a clear hierarchy in the data, that is, participants who demonstrated higher levels in the outcome space also mentioned lower levels of the hierarchy, whereas the opposite did not occur. By the end of this process I had established an identifiable set of categories of the reported experiences, with internal consistency and clear differentiation between the categories. I found they did constitute a hierarchy, wherein the higher order categories incorporated lowerorder ones with the whole forming a cohesive outcome space. In the case of conceptions of professional communication, the hierarchical levels constituted jargon and notation (the lowest level), concepts and thinking (medium), and strength (the highest level). For example, those participants who described using mathematical discourse at the highest level also described the need to avoid technical terms, whereas for several participants their only stated form of mathematical discourse was to avoid technical terms (the lowest level).
The results are described in Sections 4.2 and 4.3. This description includes what graduates do (action) and why they have selected that action (intention) and demonstrates a hierarchical set of responses to workplace situations. The outcome space includes excerpts from the interviews to illuminate the categories. The quotes are as the participants have spoken, with some deletion of hesitations and other repetitions. Spoken English is full of grammatical errors and these have not been corrected. Quotes have been shortened and sections that are not relevant to the particular context have been omitted; these are shown by the use of an ellipsis (…). At all times, care has been taken not to change the sense of what a participant is saying.
4 Results
4.1 Mathematical discourse practices
In this section, I describe the discourse practices engaged in by the graduates in their professional work and the purpose (intention) in undertaking the discourse. The macro skills of discourse are reading, writing, listening and speaking; these are tested separately in language tests such as the International English Language Testing System, which is used in Australia and elsewhere for English proficiency on entering university. In undergraduate mathematics study, the skills of reading and listening are emphasised so, although I looked at the full range of discourse skills, a summarised version is presented here, focusing on particular features of speaking and writing within the context of the graduates’ work situations.
The results of the phenomenographic analysis of the participants’ conceptions of communication are presented in Section 4.2. This includes what they do (action) and why they have selected that action (intention). Finally, in Section 4.3, I discuss the ways in which the graduates perceived they had learnt discourse skills.
4.1.1 Speaking
Presenting seminars
Boris: Somebody just said, I want to talk about something, and you have an article or something that you hand it out beforehand…And you just go over talk…and we talk about what we don’t understand, and how it affects you. But otherwise…I can always stand up, go to somebody[’s] office and start talking about the mathematical ideas when we write on the whiteboard basically.
Talking with colleagues and management
Paul: I have a hard time explaining things without being technical … I think that the majority of ideas, if you sit there and think, “how would I explain this (and this is how I think about it) to my brother”, then usually if you can work out how to explain it then it will work with my work colleagues.
Roger: Well, fortunately there [at my workplace] both of the bosses were PhDs in maths themselves so they understood. So you could just leave everything in the mathematical form, you didn’t have to first interpret it back into some handwaving thing…If we had to give reports to the nonmathematical people, then we would use, we would do examples, we would have our, all our own equations in the background but then we would [give] certain examples…
Negotiating and selling ideas
Roger: Oh, well, you must explain to them in general terms, again you have, you cannot explain it mathematically to them, that is not the point. You have to…somehow convince them that you are capable of doing this…because to them mathematics consists of arithmetic, maybe there’s an equation with an x in it that must be solved, but that’s about as far as,…everything they have done which is school maths, which is nothing in comparison with even third year mathematics.
Paul: In both my previous roles…everyone else was finance, accounting something along those lines, and weren’t maths literate so if I came up with ideas or something like that I’d have to dumb down the way I was explaining it…You’ve got to actually sell the idea as an idea.
4.1.2 Writing
Informal writing
While speaking and formal writing formed a large part of their communication, participants also mentioned informal writing; for example, I’ll just reword and then it might be a matter of just getting on to the whiteboard [Heloise]. This is an instance of informal writing, but the whiteboard was an essential component of communication. Informal writing is a powerful element of mathematical communication. Graphs, often demonstrated on computer, are also a part of informal writing used to communicate with colleagues and clients who are mathematically less skilled.
Formal writing
James: Actually I try, it’s a hard balance to be honest. For example I wrote a paper on the stability of the portfolio of deposits that we have and it’s pure stats as far as standard deviation 99% and 95% confidence intervals, for a report paper for example. So I’m using the maths in that but I wouldn’t give them the spreadsheet with all the formulas on it. I’d just say “using data back for the last 18 months I can be 99% confident that the portfolio won’t be more than 3 or 5 million dollars in a day”…you have got to get the information there but you don’t kill them with the detail. So I do use the stats to get the results but then how much of that do I use to explain the result? It depends on the audience.
4.2 Conceptions of professional communication
4.2.1 Levels of conception
Participants described how they use mathematical communication, but what is the essential variation that makes one description (experience) different from another? Using the phenomenographic paradigm to address this question and illuminate the outcome space, I discuss their responses within the context of each person’s entire transcript. I found that this outcome space primarily relates to the participants’ conceptions of discourse when dealing with nonmathematicians.
Analysis of the data resulted in three ordered categories of graduates’ conceptions of mathematical communication with level 3 being the highest level and inclusive of the lower categories.
Level 1: jargon and notation
Thi: Talking to colleagues we can go into a more technical discussion about our field and they’ll understand the lingo. Predominantly, it’s the lingo…whereas if I was to start an indepth conversation with a client, I’d have to explain to them what every word meant first.
Level 2: concepts and thinking
Melanie: You’ve somehow got to convey to somebody who thinks that maths is just a couple of numbers and equations maybe, and a tick or a cross, like something’s right or wrong…it’s kind of trying to get over that hurdle and say, OK well the maths that you know about isn’t actually what maths is.
Level 3: strength
James: You can do a bulletproof argument without being subjective, you can just quote the facts…For example if you work in a bank if you look at the issues right at the moment that are around its losses, if you have someone who is an expert in that field, they are going to be a maths person no doubt, I’m sure everyone in management is asking, “what’s going on here?”. You need to be able to communicate in general and in jargon.
4.2.2 Actions and intentions
Intention and action towards nonmathematicians
Conception  Intention  Action 

1. Jargon/notation  A. To be more efficient and friendly  A. Omit technical terms and equations 
B. To avoid losing the audience  B. Omit technical terms and equations  
C. To simplify the language to help the audience to understand  C. Avoid technical terms and repeat yourself in different ways – observing the response of your audience  
2. Concepts/thinking  A. Don’t explain—the audience will never understand  A. Do nothing—it is too difficult to explain mathematical thinking 
B. To give an impression of a mathematical concept—the audience will never understand the real idea  B. Dumb down ideas, use “hand waving” and pictures  
C. For the audience to understand  C. Careful exposition of ideas, explaining in different ways, teaching  
D. To give only an impression of a mathematical concept—the audience does not want and/or need to understand  D. Careful discourse to give an overview without detail  
E. To win a contract or an argument—the audience cannot or does not need to understand details  E. Be inspiring and sell ideas instead of explaining  
3. Strength  Justify the mathematics in an appropriate context so that the audience will understand the consequences of the mathematics. Present ideas ethically and correctly  Use mathematics and mathematical discourse in a flexible way, as the situation requires. Check for accuracy and correctness 
There is a qualitative difference between levels 1 and 2 in both intention and action, and again between levels 2 and 3 (Table 3). For the two lower level conceptions (levels 1 and 2), some participants were in a work situation where it was important for their colleagues to understand the mathematical details even though the participants modified their use of words and, perhaps, simplified the ideas. For the highestlevel conception, no participants required their audience to understand the mathematical details. They had confidence in their own mathematical knowledge and their ability to communicate it meaningfully in their workplace. They did not expect their audience to understand the mathematics, but recognised that their audience needed to be able to use the mathematical knowledge.
4.3 Learning discourse
Just as I interrogated the data to tease out graduates’ conceptions of communication (Section 4.2) using phenomenographic methods, I also analysed the interview transcripts using the same process for their conceptions of how they had learnt mathematical discourse. I found that participants expressed three levels of understanding of how they learnt mathematical discourse, with the key qualitative difference being the level of control that the graduate exercised. The first level is one where trial and error is used to work out the right communication technique for the situation. For the second level, graduates learnt communication skills from an outside source, for example from their bosses, or an outside agency such as a church group or sporting organisation. On the third level, graduates stood back, systematically observed good and bad communication, and actively modelled good practice. None of the graduates believed they had learned communication skills as part of their studies; in fact more than one graduate made a comment like: Those sort of people skills I do not think, one certainly cannot learn them at the maths department! [Roger].
In this section, I have again used quotes to illustrate the outcome space.
Level 1: go with the flow, trial and error
Interviewer: So, how did you learn to talk differently to different people?
Heloise: I don’t know. I just kind of picked it up. Mainly I’d respond by the way they spoke to me. If they speak analytically to me…with my manager…So, I know that I can speak like that to him, type thing…
Level 2: mediated by others and outside situations
Paul: Experience. After, when you first come out of uni[versity] and you try to explain … and you run it past your boss, who says it is way too technical, take all the technical stuff out of it and just give them what they can understand, write for the audience. If you hear that and get told that often enough, then sooner or later, it changes the way that you deal with people at work.
Level 3: active, detached observation
Roger: There were two kinds that stood out. Firstly those that stood out in a positive way, and those that stood out in a negative way! So then the ones that stood out in a negative way, they were exactly the sorts of people you would expect to come out of the maths department almost, very dry and techy…and then on the other hand the interesting people had a different style, you know, they spoke about interesting things…even in the context of their work, they’d always try to approach it from the more general perspective…Yeah, it was quite a learning curve I must say.
These levels are hierarchical: those with level 3 responses also used trial and error (level 1) and outside influences (level 2), and those whose experiences were mediated by others (level 2) also used trial and error. The difference between these levels is one of control. At level 3, the graduates are exercising control of their environment by using observational skills, making their own judgments and then modifying their behaviour. At level 2, the graduates are being directly moderated by their boss or outside influences; at level 1, the graduates are not in any position of control and are merely reacting to each situation in which they find themselves by using simple coping mechanisms. The participants who demonstrated a level 3 conception of learning to communicate were using critical language awareness (Jørgensen & Phillips, 2002), by observing the discourse practices of the workplace and modifying their behaviour in a much more detached and scientific way from those who used trial and error—this is what differentiates this level from the two below. It is notable that the participants believed they learnt their skills once leaving university.
In terms of the links between conceptions of mathematical discourse and learning the discourse, those who considered that mathematical discourse is jargon tended to have learnt their communication skills by trial and error or by mirroring those around them. All participants modified their use of jargon when dealing with nonmathematicians, but those with higher level conceptions used other strategies as well to enhance their communication.
5 Discussion
These graduates have come to define themselves as “mathematicians”; they see their work and life in terms of mathematics even if they are not working as mathematicians. All exhibit sophisticated conceptions of mathematics. They have assimilated their learning experiences at university and these have changed their way of looking at the world, then their jobs have given them a wider perspective again.
The study revealed that these graduates were seriously underprepared in terms of communication skills for joining the workforce. Although all participants who used mathematics in their employment were in positions where they had to communicate mathematics to nonmathematicians, they were generally unaware of language choices and how to communicate at different levels. An interesting aspect of their difficulties was the degree to which they raised problems with oral communication and interpersonal skills. Many of the mathematicians were working alone within their area of expertise and had to adjust to the language of those around them—with no training for this from university.

jargon/notation

concepts/thinking

strength
Of these, the strength conception was the most inclusive, and jargon (or notation) the least inclusive. When explaining mathematics to nonmathematicians, the firstlevel strategy is to take out the technical jargon or notation. They next tried to make the concepts more accessible to lay people. One participant described how he imagined his brother as the audience when discussing mathematical ideas in his workplace. (Is this a compliment to his brother? I think not.) Other participants described how different the way of thinking is for those trained in mathematics, and the difficulty of explaining the concepts to others. Those who could explain clearly had gained greater control and strength in the workplace. There was a qualitative difference between each level with the most marked difference being between levels 2 and 3.

go with the flow, trial and error

mediated by others and outside situations, and

active, detached observation
While not directly asked, several participants commented on the consequences of poor communication skills in the workplace, not only for themselves but also for the whole organisation. They also talked about the potential for personal problems with advancement if they were unable to adequately express details of their work practices with colleagues and managers.
There are implications for curriculum development worth considering here. I believe, as implied by Morgan (1998), that an appropriate course of action is to investigate how practising mathematicians use mathematical discourse, and develop teaching and learning strategies on this basis to induct students into the discourse of the discipline. A oneoff subject, however, is not likely to be sufficient to develop the full range of the discourse skills required. Mathematical discourse needs to be taught explicitly, not just within the context of mathematics as a discipline, but also in the context of enabling students to interact with a wider audience (some techniques are presented in Wood and Petocz (2003) and in Wood and Smith (2007)). It is, after all, ultimately the responsibility of speakers to tailor their communication to their audience.
There is a real need for curriculum reform to help our students with adjustment to the workforce, so that we do not squander people who have technical skills but lack the necessary communication and interpersonal skills. Better mathematical communication will open more employment for mathematicians as employers become more aware of the power of mathematics in their organisations.
Notes
Acknowledgements
Many thanks to the graduates who participated. Thank you to Anna Reid and Glyn Mather for their input into the study.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References
 Akerlind, G. (2005). Learning about phenomenography: Interviewing, data analysis and the qualitative research paradigm. In J. A. Bowden & P. Green (Eds.), Doing developmental phenomenography (pp. 63–74). Melbourne, Australia: RMIT University Press.Google Scholar
 Barton, B. (2008). The language of mathematics: Telling tales. New York: Springer.Google Scholar
 Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Dordrecht: Kluwer.Google Scholar
 Burton, L., & Morgan, C. (2000). Mathematicians writing. Journal for Research in Mathematics Education, 31(4), 429–453.CrossRefGoogle Scholar
 DurandGuerrier, V. (2004). Surreptitious changes in letters´ status in mathematical discourse: A source of difficulties in understanding, elaborating and controlling demonstration. In Proceedings of International Congress on Mathematics Education, ICME10, Copenhagen, July 4–11. http://www.icmeorganisers.dk/tsg25/. Accessed 20 October 2007
 Duval, R. (1999). Representation, vision and visualisation: Cognitive functions in mathematical thinking. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting North American Chapter of the International Group of PME (pp. 3–26). México: PME.Google Scholar
 Duval, R. (2001). The cognitive analysis of problems of comprehension in the learning of mathematics. Psychology of Mathematics Education, 25. http://www.math.uncc.edu/~sae/dg3/duval.pdf. Accessed 20 April 2005
 Glynn, S. M., & Muth, K. D. (1994). Reading and writing to learn science: Achieving scientific literacy. Journal of Research in Science Teaching, 31(9), 1057–1073.CrossRefGoogle Scholar
 Halliday, M. A. K. (1985). Spoken and written language. Geelong, Australia: Deakin University.Google Scholar
 Halliday, M. A. K. (1991). Writing science: Literacy and discursive power. London: The Falmer Press.Google Scholar
 Hammill, L. (2010). The interplay of text, symbols, and graphics in mathematics education. Transformative Dialogues: Teaching & Learning Journal, 3(3). http://kwantlen.ca/TD/TD.3.3/TD.3.3_Hammill_Mathematics_Education.pdf. Accessed 9 July 2010
 Hoyles, C., Wolf, A., Kent, P., & MolyneuxHodgson, S. (2002). Mathematical skills in the workplace. http://www.ioe.ac.uk/tlrp/technomaths/skills2002/. Accessed 22 August 2006
 Johnston, B. (2003). The shape of research in the field of higher education and graduate employment: Some issues. Studies in Higher Education, 28(4), 414–426.CrossRefGoogle Scholar
 Jørgensen, M., & Phillips, L. (2002). Discourse analysis as theory and method. London: SAGE Publications.Google Scholar
 Kent, P., Hoyles, C., Noss, R., & Guile, D. (2004). Technomathematical literacies in workplace activity. International seminar on learning and technology at work, Institute of Education, London. http://www.lkl.ac.uk/kscope/ltw/seminar2004/KentLTWseminarpaper.pdf. Accessed 22 December 2006
 Kvale, S. (1996). Interviews: An introduction to qualitative research interviewing. California: Sage Publishing.Google Scholar
 Marton, F. (1994). Phenomenography. In T. Hus´en, & T. N. Postlethwaite (Eds.) The international encyclopedia of education, 2nd ed., vol. 8 (pp. 4424–4429). Oxford: Pergamon.Google Scholar
 Marton, F., & Booth, S. (1997). Learning and awareness. New Jersey: Lawrence Erlbaum.Google Scholar
 Morgan, C. (1998). Mathematicians writing: The discourse of investigation. London: Falmer.Google Scholar
 Nixon, D.T., & Akerson, V.L. (2002). Building bridges: Using science as a tool to teach reading and writing. In Proceedings of the Annual International Conference of the Association for the Education of Teachers in Science (pp. 1–26).Google Scholar
 O’Halloran, K. (2005). Mathematical discourse: Language, symbolism and visual images. London: Continuum.Google Scholar
 Österholm, M. (2005). Characterizing reading comprehension of mathematical texts. Educational Studies in Mathematics, 63, 325–346.CrossRefGoogle Scholar
 Pimm, D., & Wagner, D. (2003). Investigation, mathematics education and genre: An essay review of Candia Morgan’s Writing mathematically: The discourse of investigation. Educational Studies in Mathematics, 53, 159–178.CrossRefGoogle Scholar
 Prosser, M. (2000). Using phenomenographic research methodology in the context of research in teaching and learning. In J. A. Bowden & E. Walsh (Eds.), Phenomenography (pp. 24–32). Melbourne: RMIT University Press.Google Scholar
 Selden, A., & Selden, J. (2001). Examining how mathematics is used in the workplace. Mathematics Association of America. http://www.maa.org/t_and_l/sampler/rs_6.html. Accessed 3 October 2007
 Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191–200.CrossRefGoogle Scholar
 Wood, L. N., & Petocz, P. (2003). Reading statistics. Sydney: University of Technology, Sydney.Google Scholar
 Wood, L. N., & Smith, N. F. (2007). Graduate attributes: Teaching as learning. International Journal of Mathematical Education in Science and Technology, 38(6), 715–727.CrossRefGoogle Scholar