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Graduate capabilities in mathematics: putting high level technical skills into context
Leigh Wood
Macquarie University, Australia
Introduction
What is the main aim in teaching students at university? Are universities inducting them into a discipline, or preparing them for a specific career or the general workforce? Do graduates require only discipline-specific skills, or are there extra capabilities they need to develop? There has been a recent emphasis in research on tertiary education about the development of graduate skills and attributes, features of successful graduates and of unemployed graduates, how teaching and learning influence the development of graduate attributes, and conversely how the concept of work influences learning. This kind of research has helped to transform university policy, which has flowed on to changes in curriculum in many institutions.
As an example of such change, Franklin (2005) describes the establishment and content of a professional issues and ethics course in a university school of mathematics, founded because of concern across the university that, the concentration of undergraduate education on technical content was not preparing students for the workplace. (p. 98). The course begins with an emphasis on graduate attributes required by employers, such as communications skills, teamwork and ethical behaviour. There are guest speakers from industry to discuss requirements in a workplace: Students soon to graduate learn something genuine about what they face (p. 99). The subject gives an overview of mathematics and its uses. General ethical issues are presented, such as confidentiality and duty of care, with the use of case studies. In support of developing such a course at other universities, Franklin suggests that, They need a break on something that connects them with what they will do when we let them out. (p. 100).
Wood and Reid (2005) also note that, Adjusting to the workforce can be problematic for students as they discover that what they have learned at university needs to be contextualized for work (p. 125). The paper focuses on the experience of communication skills, which are used as an indicator of the success of the transition to the workplace. These are likely to be different for a student and a professional mathematician, a point that has had, insufficient attention paid to [it] within curriculum design at university. (p. 126). The authors recognise that communication within mathematics can be problematic for students, but assert that a greater difficulty lies in determining, how practicing mathematicians use mathematical discourse (p. 126) in order to, develop teaching and learning strategies to inculcate students into the discourse of the discipline. (p. 126).
The research revealed that no graduate believed that they had studied mathematical communication at university and graduates had robust criticism of their preparedness for the workforce and the process of getting a job. The authors recommend that universities should have a compulsory professional development subject for all students (as in Franklin, 2005). Other changes are recommended and it is emphasised that most work should be done within the teaching and learning of the technical subjects themselves due to the discipline-specific nature of the language and communications required.
Petocz and Reid (2006) conferred with recent graduates too, and found that studying mathematics was perceived by the participants as having contributed to their ability to solve problems and think logically: all our respondents put forward the idea that studying maths had helped them in the general area of problem solving ... (p.4). They link the graduates interpretation of these skills with the graduate views of being a professional, using the notion of the Professional Entity previously developed by the authors.
Graduate attributes are also important in the learning and teaching of service mathematics (Mills & Sullivan, 2005). The area of graduate attributes is described (such as for engineers) in regard to how mathematics can contribute to these. They make suggestions as to how professional requirements can be built into teaching, with a few examples (such as how a question is posed in mathematics). They also make it clear that research should inform teaching since these are the functions of universities so, it is important that undergraduate students get a taste of both throughout their training. (p. 84).
Selden & Selden (2001) summarised studies on how much mathematics is actually used in particular occupations. An interesting finding was the different ways biologists and statisticians viewed modelling: The mathematicians thought in terms of mimicking the situation, ie., providing a model that is both descriptive and predictive The biologists, on the other hand, felt that biological systems are far too complex for this They saw the role of models as stimulating conjectures, ruling out possibilities, and serving as experiments for theoretical claims. (p. 4).
A UK investigation into using mathematics (Hoyles et al., 2002) makes the point that application of known mathematics can be complicated in the workplace: the need for workers to recognise immediately that a particular piece of mathematical information was implausible and possibly just wrong. This kind of judgement requires a combined appreciation of mathematics and the context and content of the work practice. (p. 12) This has implications for university teaching: It is important that employers and education and training providers recognise this fundamental fact. educators should recognise the importance of giving students experience with multi-step, data-dependent operations. (p.12) FitzSimonss research (2006) also led to her belief that it is essential that curriculum and pedagogy are oriented towards the outside world, and that there should be some integration with potential work experience.
Graduate capabilities
There is some confusion about terminology here. Are graduates developing skills, attributes, capabilities, dispositions or all of these? Are these generic or discipline specific?
Bowden et al. (2000) consider generic graduate attributes to be:
The qualities, skills and understandings a university community agrees its students should develop during their time with the institution. These attributes include but go beyond the discipline expertise or technical knowledge that has traditionally formed the core of most university courses. They are qualities that also prepare graduates as agents of social good in an unknown future.
To identify the generic achievements and get agreement on them by stakeholders is not easy. A large Australian study reported by Hambur, Rowe and Luc (2002) tested graduates over a range of graduate skills. They selected 5 cognitive dimensions to assess. Critical thinking, problem solving and interpersonal understandings were each tested using 30 multiple-choice items; argument writing and report writing were assessed using a writing task. The items were changed for context in different disciplines. These cognitive dimensions were selected after consultation with universities and other stakeholders such as employer groups and professional bodies. Employers preferred skills that helped their organisations with their goals, especially personal and interpersonal skills, while the universities focused more on academic skills and qualities related to citizenship. Major findings included those that may be expected, for example, Arts/Humanities students performed better on critical thinking and interpersonal understandings, whereas Engineering and Architecture students did relatively better on problem solving.
The report is an important contribution to the discussion on graduate skills. Mathematics and Science students (grouped for the report) perform around average for all domains, slightly higher for problem solving and slightly lower for argument writing, and with less variability than students in other domains. Another useful finding was that scores on the domains tested were significantly higher in later years of study. The authors express caution concerning this result, as the reasons for it are not clear. However, if the finding is correct, and students are improving on these graduate skills, then this is a positive result for teaching and learning.
Other recent research focuses on successful graduates (Scott & Yates, 2002; Scott, 2003). For a particular field of study, several employers were selected and asked to nominate a group of their most successful recent graduates, about 20 in all. The graduates and supervisors were then interviewed in depth to ascertain the attributes that had contributed to the graduates success. From his research, Scott has developed a framework of professional capability (Scott, 2003, p. 5). Scotts research points out that it is when things go wrong that professional capability is most tested, not when things are running smoothly or routinely. It is at times like these that the individual must use the combination of a well-developed emotional stance and an astute way of thinking to read the situation and, from this, to figure out (match) a suitable strategy for addressing it, a strategy which brings together and delivers the generic and job-specific skills and knowledge most appropriate to the situation. All the research leads to the conclusion that, while technical expertise is a necessary capability it is certainly not sufficient, to produce a successful graduate (Scott & Yates, 2002).
Several professional societies have grappled with the development of graduates, in particular in the area of ethics and professional responsibility. Engineers Australia and the Statistical Society of Australia both have codes of conduct for their members. Engineers Australia has a graduate program where they consider the professional education of their graduates. Graduates have a mentor and development program that emphasises the competence and responsibility of an engineer. This approach could be beneficially used in other professional areas including science disciplines.
Research design
The major focus of the study reported here was to examine the transition to the workforce for mathematics graduates, with communication skills being used as an indicator for the ease of transition. The mathematics and discourse used by new mathematics graduates in industry was examined alongside their perceptions of how they acquired these skills. I analysed interviews with these graduates and texts developed for work purposes to illuminate their perceptions of how they use this discourse and how they learnt their communication skills. Mathematical discourse refers here to the uses of language in university mathematics learning and teaching as well as in professional life (Wood & Perrett, 1997, p. i).
The participants had finished their studies in the past five years and they had a range of employment situations, age, ethnicity and academic backgrounds. The 18 participants had graduated from five universities and had a range of degrees. Most of the participants were in their early to mid twenties. There were three people who had studied mathematics as their second degree and one had come to university as a mature-aged student. There were 10 males and 8 females; 8 spoke languages other than English as their home language. The commonality between them is that they all regarded themselves as qualified mathematicians. It should be noted that all names have been changed and ethics approval was obtained from Macquarie University for the research.
The themes that emerged from the transcript group as a whole comprised: getting a job; initial work experiences; use of mathematics; working as a mathematician; lack of communication skills; and changes to university programs. To illustrate the themes I use a number of quotations that emphasise the complexity and variation apparent in each graduates job.
Results of the study
Use of mathematics
How do these graduates use mathematics in their early careers? The responses range from bugger all through the most simple mathematics equations to sophisticated mathematical modelling. All graduates felt that they knew more mathematics than was required for their positions, though this was not seen as a negative since it gave them confidence and room to move. Most used their textbooks when they had forgotten details of the mathematics or needed to develop new work.
Leah: It [statistics] just gives you a lot of points of view to start looking at problems, whereas statistics is very much about, Who says? Wheres the evidence? What are you basing that on?, Is there a real trend?, Is it just a one-off?, Back it up.
Roger: I think the hardest thing that I ever had to actually use, in my job was second year maths, these Riemann sums, that was about the hardest. But it really helped to know a lot more, because you can more freely work with the things.
Paul: A couple of weeks ago I was given a new swaption, this weird derivative to price, and how do you do it? And I kind of sat down and had a think about it and worked through, pulled out an old derivatives securities textbook and worked through it like a problem, that was it.
Gavin: [Im] working with a climate modelling group on what they call data assimilation, which is ways of incorporating observations, real observations to better kind of models, I guess, so, using neural networks, using multi-dimensional calculus stuff, using, so its sort of applied maths really.
Boris: Im actually doing cipher design, to be able to design that, you need the mathematical theory to prove security bounds.
Graduates wanted more exposure to real-world situations as part of their learning:
William: a graduated approach, where you might start by learning some theory, then be working a bit with, say, a lecturer in a mock team situation on a realistic project, then having industry experts come and work with you to maybe do real project So, I suppose, if university was able to forge stronger co-operative project-based work with industry, that would be a really helpful thing to make that transition.
There were several references to content in university programs. The majority of these were to do with specific computer products, such as Excel, Visual Basic or SAS (a statistical package). The following quote is typical:
James: As far as transition for work, everywhere uses standard products like Excel, and if you come out of a maths degree, I wasnt really taught to use Excel all that much here [at university] and I think its really a tool of the trade.
The tables below demonstrate the range of mathematics and computing skills that these graduates actually used in the workplace.
Table 1 Level of mathematics used in the workplace
0 None
University mathematics study not required1 Low
First year university level mathematics2 Medium
Second year 3 High
Third year or higherChristine
Leah
Melanie
WilliamAngie
Fredrik
Nathan
ThiDavid
Evan
Heloise
James
Sally
RogerBoris
Gavin
Kay
Paul
Table 2 Level of computing used in the workplace
0 General1 Standard tools (Spreadsheet, database)2 Specialist (VB, Mathematica, SAS)3 Programming (various high-level languages)Christine
Leah
MelanieAngie
David
Heloise
James
PaulFredrik
Gavin
Kay
Sally
ThiBoris
Evan
Nathan
Roger
William
These graduates are using mathematics in different ways. The ones who are not explicitly using the mathematical procedures that they learnt at university nevertheless believe that they have taken on the characteristics of a mathematical person, such as logical thinking and being more aware of numerical and logical situations around them. They have a mathematical identity.
Working as a mathematician
Many mathematics graduates encountered a number of difficulties out in the workplace, for instance some found that their bosses may be demanding results that are simply not possible from the mathematics or the data available. Roger, in particular, expressed serious frustration with the way mathematics was used in his workplace and had to change his ideas of working as a mathematician. He was incredulous about the demands made on him (and mathematics) and the way he was expected to work. Both Evan and Roger talk about having to change your ideas about mathematics and how it is used in the real world, to relax your assumptions. Evan says: this is theory and this is what really happens. He is making connections between the perfect mathematics and financial theory he has learnt in the classroom and the reality of dealing with real financial situations.
Roger: So, basically, there were many absurdities of this nature where they expected you from minimal data to extract more information than the data could provide. So that was quite irritating. And there was no stopping them demanding it. One problem for people with pure maths, well that I certainly had, is it is quite a shock to them in the real world to see how maths is used in this strange way where assumptions are made left and right, whereas in pure maths one dare not.
Evan: Maths has a whole heap of simplifying assumptions that say, you know, this is what youve, these are all your assumptions, and theyre all fixed or theyre all ah, perfect-world constraints And so its taking those compromises into the real world, and I guess seeing all these pure maths and saying, OK, now these are the sort of things that we need to relax in a real-world context, and these are the sort of ah, this is theory and this is what really happens And so thats very important to get that understanding.
Paul: I guess what Im using is the statistics side of what I did at uni and also some of the financial maths that I did so the derivatives pricing and things like that. Coming up theres going to be a project where Im going to get into the stochastical modelling, basically because rather than using deterministic risk models were going to use stochastic ones.
Using mathematics to communicate ideas
This is where interesting differences between graduates emerge. In other parts of the interview, graduates are in familiar territory. They are describing how they got their jobs, their initial experiences and how they use mathematics. The majority of the graduates had not considered the use of mathematics to communicate ideas or that they used mathematics to communicate ideas or that they were communicating at all.
Boris: I can always stand up, go to somebody[s] office and start talking about the mathematical ideas when we write on the whiteboard basically.
Gavin: Im working by myself, so when I communicate with my supervisors, its broad brushstrokes.
Heloise: The sales manager, shes from a sales background, so not as analytical as myself and my manager and a lot of times when I talk to her, I just try not to be as mathematical, or as analytical, about something. The good thing is shes very open, shell just say, English please! Cause she knows that we dont mean it, even when my manager talks to, we dont mean thats just how we talk. Shes like, hang on, just run that past me again!
Graduates move into a wide variety of work situations. The majority of the participants were the only people in their workplace who have qualifications in mathematics and who could speak the mathematical language. It is clear that, within their workplaces, these graduates have a mathematical identity but do not form a community of practice. They are modifying their practices to fit in with the dominant workplace situation. Communication skills are seen as an important part of preserving the job and progressing in the workplace.
Conclusions
Most students who study mathematics at university do not proceed to gain a major in the mathematical sciences. Those who gain a mathematics major generally do not go on to become academic mathematicians. For these graduates, the basis of this investigation, the amount of technical mathematics that they use in their workplaces is less than what they have studied in the three years at university. Nonetheless, they have taken on a mathematical identity and have gained confidence and a range of problem-solving skills that are transferable to their work and personal lives.
This study has shown that mathematics learning and teaching at university are not preparing students for professional life after university. Many graduates are unable to release the strength of their mathematics because they have not been taught how to communicate mathematically. Many are not given the opportunity because of the low level of mathematics required for their work. Some are unable to adapt to the work environment. Their technical mathematics skills are high but many did not use these skills or had to learn others. Graduates required more computing skills than were taught as part of their degrees.
Participants have suggested changes to content, learning methods and structure of university programs to assist with their transition to the workplace. These results complement studies of employer mathematical needs by Kent et al. (2004), which found an increase in the requirement for mathematical literacy amongst employers and for skills of mathematical communication in the workforce.
The fact that these graduates did not demonstrate a broad knowledge of mathematics (in fact, they demonstrated the opposite) or the idea that they could develop new mathematics suggests that these are serious areas of omission in the curriculum. Even if students are going to specialise in an area of mathematics, they should be exposed to the breadth of their subject at an early stage.
In terms of curriculum design, we need to consider the whole learning experience of undergraduates, not just what is directly taught and assessed. An integrated approach to course design across an undergraduate program seems sensible, but one of the main barriers to the introduction of explicit teaching of graduate attributes is historical. While it is important that graduates are able to perform mathematical and computing techniques and know the relevant jargon and notations, it is essential that they are able to communicate their knowledge in a variety of circumstances and work in multidisciplinary teams in the workplace. This is achieved through modelling an extensive range of professional activities using authentic materials.
References
Bowden, J., Hart, G., King, B., Trigwell, K. & Watts, O. (2000). Generic capabilities of ATN university graduates. Canberra: Australian Government Department of Education, Training and Youth Affairs. Retrieved from HYPERLINK "http://www.clt.uts.edu.au/ATN.grad.cap.project.index.html" www.clt.uts.edu.au/ATN.grad.cap.project.index.html [April 16, 2008].
FitzSimons, G.E. (2006). An activity theory perspective on technology mediated mathematics education for undergraduates. In Proceedings of the 3rd International Conference on the Teaching of Mathematics at the Undergraduate Level. Istanbul, Turkey: Turkish Mathematical Society.
Franklin, J. (2005). A Profession Issues and Ethics in Mathematics course. Australian Mathematical Society Gazette, 32, 2, pp. 98-100.
Hambur, S., Rowe, K. & Luc, L.T. (2002). Graduate skills assessment. Commonwealth of Australia. Retrieved from HYPERLINK "http://www.dest.gov.au/sectors/higher_education/publications_resources/other_publications/graduate_skills_assessment.htm [2" http://www.dest.gov.au/sectors/higher_education/publications_resources/other_publications/graduate_skills_assessment.htm [April 2, 2008].
Hoyles, C., Wolf, A., Molyneux-Hodgson, S. & Kent, P. (2002). Mathematical skills in the workplace. London: Institute of Education - Science, Technology and Mathematics Council.
Kent, P., Hoyles, C., Noss, R., & Guile, D. (2004). Techno-mathematical literacies in workplace activity. International seminar on Learning and technology at Work, Institute of Education, London. Retrieved from http://www.lkl.ac.uk/kscope/ltw/seminar2004/Kent-LTW-seminar-paper.pdf [December 22, 2006].
Mills, T.M. & Sullivan, P. (2005). Mathematics at your service. In Ko, H.K. & Arganbright, D (Eds.), Proceedings of KAIST International Symposium on Enhancing University Mathematics Teaching (pp. 81 95). Korea: KAIST.
Petocz, P., & Reid, A. (2006). The contribution of mathematics to graduates professional working life. In Jeffery, P.L. (Ed.) Australian Association for Research in Education 2005 Conference Papers. Retrieved from http://www.aare.edu.au/05pap/pet05141.pdf [July 18, 2007].
Scott, G. & Yates, K.W. (2002). Using successful graduates to improve the quality of undergraduate engineering programs. European Journal of Engineering Education, 27, 4, 363378.
Scott, G. (2003). Using successful graduates to improve the quality and assessment in nurse education. Proceedings of the Australasian Nurse Educators Conference. Retrieved from HYPERLINK "http://www.uws.edu.au/ download.php?file_id=8571&filename=SuccGrads_NsgEd__GS_Final03.pdf&mimetype=application/pdf" http://www.uws.edu.au/ download.php?file_id=8571&filename=SuccGrads_NsgEd__GS_Final03.pdf&mimetype=application/pdf [December 22, 2006].
Selden, A. & Selden. J. (2001). Research Sampler 6: Examining how mathematics is used in the workplace. MAA Online. Retrieved from HYPERLINK "http://www.maa.org/t_and_l/samplers/rs_6.html" http://www.maa.org/t_and_l/samplers/rs_6.html [October 3, 2007].
Wood, L. N. & Perrett, G. (1997). Advanced Mathematical Discourse. Sydney: University of Technology.
Wood., L. & Reid, A. (2005a). Mathematics communication for graduates. In Bulmer, M., MacGillivray, H. & Varsavsky, C. (Eds.), Proceedings of Kingfisher Delta 05 (Fifth Southern Hemisphere Conference on Undergraduate Mathematics and Statistics Teaching and Learning, pp. 125-131). Brisbane: University of Queensland.
Wood, L. & Reid, A. (2005b). Graduates initial experiences of work. In Jeffrey, P.L. (Ed.), AARE 2004 Conference Papers Collection. Melbourne, Victoria: The Australian Association for Research in Education. Retrieved from HYPERLINK "http://www/aare.edu.au/05pap/woo05147.pdf" http://www/aare.edu.au/05pap/woo05147.pdf [April 16, 2008].
Wood L. & Reid, A. (2006). Conversations with graduates: Reflections on learning mathematics. In Proceedings of the 3rd International Conference on the Teaching of Mathematics at the Undergraduate Level. Istanbul, Turkey: Turkish Mathematical Society. Retrieved from HYPERLINK "http://www.tmd.org.tr/ictm3/" http://www.tmd.org.tr/ictm3/ [April 7, 2008].
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.
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