Last modified 14 July 1998. Final Version.
Mathematics in New Zealand: Past, Present and Future.
Report No. 77 ISSN 11710101 July 1998
Published by the Ministry of Research, Science and Technology PO Box 5336, Wellington, New Zealand Telephone: +64 4 472 6400 Fax: +64 4 471 1284 Internet: http://www.morst.govt.nz
Approved for general release Dr James Buwalda Chief Executive
During the February 1996 meeting of the Mathematical and Information Sciences Standing Committee (MISC) of Royal Society of New Zealand, the Committee met with Professor Don McGregor, the Chief Scientist of the Ministry of Research, Science and Technology (MoRST). Following discussions it became clear that the role of the mathematical sciences in the general sciences scene in New Zealand was facing difficulties. It was agreed that Professor Graeme Wake, the Chair of MISC, would write a scoping paper which would outline the areas of concern. This was sent to the Ministry of Research, Science and Technology for consideration towards the undertaking of a review by MoRST.
In April 1997 the Ministry submitted to the then Chair of MISC, Professor Jeffrey Hunter a draft Terms of Reference for a Mathematics Study with a request that a report be submitted by 31 July 1997. These Terms of Reference were similar to those put in place for the Australian Review which culminated in the Report "Mathematical Sciences – Adding to Australia" in January 1996. Professor Hunter met with Professors Marston Conder, Graeme Wake, David Gauld and Alastair Scott during a visit to Auckland in early April and consulted with Professor David VereJones. It was clear from these discussions that it would be impossible to carry out the task in the time frame requested.
Early in May 1997 MoRST agreed to split the exercise into two areas: the first (subcontracted to Mr Malcolm Menzies, Victoria Link) to be completed by the end of July, a review of the Public Good Science Fund (PGSF) proposals with the aim of determining the extent of the underpinning applied mathematics (including Statistics) input, and the second to commence as part of the knowledge foresight exercise involving their priority setting for the PGSF. The second part was to be split into two phases – scoping requirements for the knowledge foresight exercise and it's completion.
This revised procedure met with acceptance of MISC and the contract was signed in early June for phase one which was to be completed by the end of July 1997, and early August for phase two due for completion by the end of May 1998.
This entire project has been a major undertaking by the mathematical sciences community. It was obvious to all involved that the task needed to be done well, that full consultation with all component disciplines was required and that the Ministry's expectations needed to be met. To this end, a small review team ("secretariat") was put together comprising Professor Jeff Hunter (chair), Professor David VereJones, Associate Professor Stephen Haslett, Mrs Jean Thompson and Dr Mark Bebbington. Meetings of this team were held on a regular basis.
The entire project would have faltered if this team had not been in place. While I was prepared to chair and coordinate the review it would have been an impossible task without the dedicated support of this team. Besides being a sounding board they all made significant contributions.
At the beginning of phase one, the author of the Australian Review report, Dr Noel Barton, agreed to visit Auckland and Wellington to meet with representatives from various mathematical science disciplines and to brief us on the procedures used in the Australian exercise. Dr Barton's clear and focussed presentation enabled us to get a good handle on the logistics of the entire process.
In July we produced a discussion document (since peer reviewed and updated as Appendix 1 to the report) on the future likely developments in various mathematical science areas. I wish to express my sincere thanks to the subject area coordinators who were instrumental in facilitating the compilation of this document: Professor Marston Conder, Professor Rob Goldblatt, Professors Douglas Bridges, David Gauld, Professor Graeme Wake, Dr Graham Weir, Professor John Butcher, Associate Professor Andy Philpott, Professor Jeff Hunter, Dr Mark Bebbington, and Professor Bryan Manly.
For the preparation of the final version the above team were complemented by Dr Harold Henderson, Mr Mike Doherty, Dr Katrina Sharples, Ms Elisabeth Wells, Dr Ray Brownrigg, Professor Les Oxley, and Mr Leigh Roberts in order to flesh out the reports in more detail. In addition the authors of the initial scenarios sought peer reviews to ensure that the final version gave a comprehensive view of future likely developments.
The contract called for questionnaires to be used to obtain responses from a wide variety of sources. These questionnaires were initially based on those used in the Australian Review. However as the Review team adapted them, it became clear that substantial refinements were necessary in order to seek information relevant to this study and much more effort was required. The Review team was augmented by Dr Michael Carter, Mr Greg Arnold and Mr Duncan Hedderley of Massey University and drafts of the five different questionnaires (Universities, Polytechnics, Research Organisations, Professional Associations and User Groups) were critiqued by the subject area coordinators and Dr Gerald Rys of the Ministry of Research Science and Technology. The construction of the sampling frame was facilitated with inputs from MoRST, the RSNZ, mailing lists of NZMS, NZSA, and ORSNZ, as well as names suggested by the subject area coordinators and other individuals including Ms Sharleen Forbes (Statistics New Zealand) and Associate Professor Kevin Broughan (University of Waikato).
The questionnaires were mailed by late September in order to give sufficient time for responses to be received prior to regional workshop meetings. These meetings were held in Auckland (4 November), Hamilton (5 November), Wellington (7 November), Dunedin (10 November), Christchurch (11 November) and Palmerston North (13 November). The local arrangements were coordinated respectively by Professor Graeme Wake, Professor Douglas Bridges, Dr Graham Weir, Professor Bryan Manly, Associate Professor Rick Beatson and Associate Professor Stephen Haslett. Professor Hunter conducted each of the workshops ably supported by Associate Professor Stephen Haslett on the northern circuit and Dr Mark Bebbington on the southern leg. The morning sessions focussed primarily on issues impacting on the tertiary sectors and the afternoon sessions looked at issues of interest to research and user organisations. Some preliminary observations from the questionnaire responses were also presented. These meetings gave the participants ample opportunities to comment and stress issues that they wished to be addressed in the final report.
Following a meeting of the review team to digest the information collected, a final workshop was held during the afternoon of December 10 at Science House, Royal Society of New Zealand, preceded by the meeting of the Mathematics and Information Sciences Standing Committee of the RSNZ.
With the meetings concluded, further detailed analysis of the questionnaire responses was necessary. The information provided in Appendix 2 of the report has been compiled from the questionnaires by Mrs Jean Thompson assisted by Dr Don Thompson and Mr Mike Camden (Polytechnics section). This analysis has been crucial to the Review team in ensuring that the comments and input reflects the views of the mathematical sciences community. We are particularly grateful to respondents who permitted their comments to be quoted in the Review.
I wish to express a debt of gratitude to Professor David VereJones who has borne the brunt of the writing of the final report. Without his efforts and dedication this project would have faltered. At the same time I wish to add that this has been very much a team effort. Secretarial help from Ms Paula McMillan (Massey University) and Ms Edith Hodgen (Victoria University of Wellington) has enabled us to complete the project within the time frame assigned.
Submissions from John Maindonald and Brian Easton brought important issues to the Review team's attention. There are others who have assisted, especially those who attended the public meetings, and made major contributions. My apologies for omitting names. However I wish to publicly record my grateful and sincere thanks to a dedicated review team who have worked tirelessly, most of it on a voluntary basis, to ensure that the views of the mathematical sciences community are faithfully recorded and communicated to the Ministry. To have been able to identify areas of concern and opportunities has been our contribution. We sincerely hope that the views of this report will be comprehended and that efforts will be taken, by those able to effect change, to ensure that the vitality of our disciplines is maintained and supported in the future.
Jeffrey J Hunter Chair Mathematical And Information Sciences Standing Committee, RSNZBack to Table of Contents
This Report is the third in a series of three studies commissioned by the Ministry of Research, Science and Technology (MoRST) to elucidate the role played by the Mathematical Sciences in New Zealand and their possible future contributions to different socioeconomic sectors. The first study in the series was devoted to an assessment of New Zealand's current knowledge base in the Mathematical Sciences. It forms part of a wider knowledgebase exercise carried out by the Ministry. The second study reported on "the underpinning requirements of mathematics disciplines in relation to the socioeconomic framework used for Government science within the Public Good Science Fund (PGSF)". It examined in particular the mathematical components of recent Foundation for Research Science and Technology (FRST) proposals in the different output classes, and some concerns of the mathematics discipline groups over the effect of current government strategies on New Zealand's mathematics capability.
The present study examines new developments in the subjectarea itself, the uses of the mathematical sciences in public and private sector organisations in New Zealand, the financial benefits of these uses, problems affecting the disciplines, and the training of mathematicians in New Zealand. From these components it attempts to put forward an integrated picture of the present state of the mathematical sciences in New Zealand, their role in the economy, and the factors likely to affect their role in the future.
To assist it in developing this picture, the review team commissioned a series of short reports on current and future developments in different fields of mathematics, first from subject experts within the universities, then from mathematicians working in different sectors outside the universities. Each report is formatted along the lines suggested for the Foresight Project in the Terms of Reference; they are collected together in Appendix 1 of the Report. Together they broaden and supplement the earlier knowledgebase exercise, particularly in respect of uses of mathematics outside the universities. The review team also developed a set of questionnaires and circulated them to users and providers of mathematics in New Zealand. The questionnaires were developed from questionnaires used in the recent Australian review, but reshaped to fit the New Zealand context and to reflect the more general objectives of the New Zealand Report. A summary of the questionnaires and an analysis of the responses to them is contained in Appendix 2 of the Report.
The initial discussions concerning this Report preceded the initiation of the Ministry's Foresight Project; the Report itself has evolved alongside the Foresight Project and has changed its focus as the review team became more familiar with the Project's own scope and objectives. In its present form, the Report attempts to link the requirements of a sectorbased, outputdriven funding programme, to a disciplinebased view of the subject's current situation and potential contributions, and merge both with the general objectives of the Foresight Project.
There are particular difficulties in accommodating research and development in the mathematical sciences to a funding programme based on outputs in specified socioeconomic sectors, because it is the very nature and strength of mathematics that its abstract character makes it contextfree. No other subject shares this feature to the same degree, nor, with it, mathematics' underpinning role.
Faced with the need to reconcile these different viewpoints, the review team's first priority has been to meet its Terms of Reference as set out in the subsection below. At the same time it has been conscious of the fact that the Ministry has provided the mathematical sciences in New Zealand with a unique opportunity for an indepth report of their current state and their possible future evolution. The review team, therefore, has also felt an obligation to its colleagues to ensure that the report gives a full coverage of the aspects that will be important for them in their own future planning. In particular, the Report comes at a time when the university departments of mathematics are faced with the need for a serious reappraisal of their role, and a special effort has been made to deal in depth with the university programme. Finally, and at a relatively late stage in its activities, the review team was kindly given access to material from the Ministry's Foresight Project Workshop, and has endeavoured to use this in linking the format and conclusions of the Report to the perspectives of the Foresight Project.
Throughout the Report, mathematics, or more generally the mathematical sciences, is interpreted in a broad sense. In particular it includes not only the different subfields of what is conventionally described as "pure mathematics" (a phrase the review team has had to approach with some care, as experience has shown that virtually no field of mathematics, no matter how abstract or esoteric, has yet emerged which has not been found to have some reallife application, sometimes of great practical or economic importance) but also statistics and operations research. There was, however, a deliberate decision made not to attempt a systematic evaluation of the roles and potential contributions to New Zealand of economics in general (although some mention of mathematical economics and econometrics is given) or of computer science and computing (although here again the links with the mathematical sciences are so tight that some discussion is inevitable) or of engineering (traditionally the largest user of mathematical techniques). Any one of these fields would require separate evaluation in their own right. A more detailed itemisation was used to orient the questionnaires, and is reproduced in Appendix 2.
In the body of the report, we have tried as far as possible to treat the mathematical sciences as a whole, rather than to launch into separate discussions of each branch. In Appendix 1, however, an attempt is made to examine the fields individually, albeit in condensed form.
Through a foresight exercise, prepare a report on future likely developments in mathematical sciences in New Zealand and internationally, and assess their impacts on
The report should:
For the field of mathematics identify
Issues such as links with interdisciplinary science and international linkages should be considered.
Development of the report should involve consultation with key providers, users and funders.
The study should also fully assess currently available information and analysis relating to mathematics foresight in order to avoid duplicating effort. Bibliometric and other quantitative analysis should be used where appropriate.
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Mathematics underpins the operation of a modern society at three levels and in three somewhat different ways.
This is perhaps the deepest sense in which mathematics underpins modern society.
Within New Zealand, as we shall explore in more depth in later sections, mathematics is a major component of the primary and secondary school programmes, is taught to graduate level in all the major universities in New Zealand, is used extensively in government and private sector organisations, and throughout the research institutes. However, despite the manifest contributions which the mathematical sciences make to the New Zealand economy, and the frequent protestations on the part of government concerning the importance of the quantitative sciences to New Zealand, there is widespread unease within the mathematical disciplines themselves that all is not as well as might be wished. The last decade has seen a substantial reduction in the size of mathematical groups, as well as in the overall number of mathematics professionals, employed outside the universities. University mathematical science enrolments at thirdyear level and higher are at best static, despite huge increases in tertiary enrolments overall. There are continuing problems with the school programmes with mathematics, as evinced by the continuing shortage of secondary mathematics teachers, and a decline in the performance of New Zealand schoolchildren relative to OECD norms.
A disturbing theme which recurs at several places in this report is a kind of national prejudice against the use of rational analysis and explicit quantitative methods. They are to be used, so to speak, only as a method of last resort. This tendency may be seen in such diverse features as the lack of representation of engineers and scientists on New Zealand Boards of Directors, the reluctance to introduce probability statements into New Zealand weather forecasts, the limited use (by international standards) of operations research techniques in New Zealand industry, the poor rewards for mathematical careers, and the poor quality of quantitative analysis in much current decisionmaking. Unless this prejudice can be overcome, it seems likely to have a greater impact on the success of national strategies based on the goal of a "KnowledgeBased Society" than any developments in the mathematical sciences themselves.
To help bring this last issue into perspective, we set out in the following subsection a scenario for the fuller role that the mathematical sciences in New Zealand might play, if some of the current obstacles were removed. In the later sections of the Report, we return to the current situation in New Zealand and to consideration of a national strategy which might help New Zealand to move from the current situation to one closer to that outlined below.
At numerous places in the Foresight Project documents, and most explicitly in the Scenarios for the Future included in the briefing material for the Foresight Project Workshop, reference is made to the desirability of New Zealand moving towards a "knowledgebased" society. For example, in a recent Notice setting out the Government's policies for Public Good Science the Minister outlined his vision of New Zealand's future as a KnowledgeBased society. "We need to be adept at both creating new scientific and technological knowledge and at learning by adapting existing technologies for innovative purposes. Knowledge creation, technological learning and innovation thrive in an environment where researchers, technologists, designers, engineers and marketers all interact routinely, share common goals, and have effective ways of communicating with teach other." Where do the mathematical sciences fit into this picture?
The basic features of their underpinning role, as summarised in the previous subsection, are hardly likely to alter. What is capable of being altered is the quality and effectiveness with which the different levels indicated there are implemented. This will determine the contribution the mathematical sciences make to New Zealand's development and future wellbeing, including its international competitiveness. We set out below some suggestions  our own Scenario for the Future  as to how, in a small, knowledgebased society, the mathematical sciences might operate.
One of the key tenets of the "Total Quality Management" (TQM) movement is that improvements in quality come from bottom up and not from top down. If the movement to higher quality is to extend into quantitative aspects, then the general workforce needs a good grasp of the mathematical tools that it is using, extending beyond the ability to follow specific procedures into an understanding of the principles on which the procedures are based. A high level of quantitative literacy is also important to the general public in its exercise of basic democratic functions  discriminating, with a proper understanding of the quantitative issues, between alternative policies, whether national, local, or at the level of community functions. The qualities to be looked for are not so much specific mathematical skills as the ability to reason clearly, and the ability to reason in quantitative terms. To an extent which remains controversial, these abilities are innate and vary from individual to individual. What is generally agreed is that the ability of any given individual to reason in these ways can be considerably enhanced by appropriate training and experience. In the knowledgebased society, a fundamental task of the school programme is seen as being to develop these abilities. High quality teachers are attracted into this work by their sense of its importance, and the recognition given to their work by the community, as manifested in salary scales, good working conditions, and social status.
Similar comments apply at the next level up. Within a knowledgebased society, the users of quantitative techniques in the various technical and professional fields have a sufficient depth of knowledge of the quantitative techniques they use (including software packages) to be aware of the limitations of the techniques, and of how they can be modified in nonstandard situations. Moreover there is good communication between the users and the applied mathematicians whose professional role it is to advise on such extensions and modifications.
To achieve this depth of understanding, tertiarylevel graduates in all fields, including commercial and social science subjects, have a depth of knowledge of mathematical techniques that is commensurate with an informed use of the procedures they are likely to encounter in their professional life. This includes in particular basic understanding of the modelling concepts involved in relevant software packages and of how the software works. This is a task the tertiary institutions take very seriously. They provide not only introductory service courses, but specialised practical classes, run partly by mathematicians and partly by subject specialists, in which these background concepts are developed, and where students with aptitude are encouraged to explore further the mathematical issues. In addition, the tertiary institutions provide opportunities for upskilling graduates in quantitative methods and enabling them to become familiar with recent developments in improved software and modelling procedures within their respective fields.
As a consequence of their own appreciation of the benefits in efficiency and informed decisionmaking to be gained from proper use of quantitative methods, managers in the knowledgebased society encourage such upskilling, and reward the ability to use quantitative methods effectively by promotion and enhanced career prospects. What they look for are the abilities to bring quantitative tools to bear on practical problems, to analyse existing procedures and problems in quantitative terms, to introduce innovations and improvements to existing procedures, to analyse clearly and communicate the benefits of alternative procedures. Ultimately, staff with such skills can expect to move into senior management and directorship positions. In turn, these opportunities encourage able highschool graduates to look towards careers in quantitative fields, and to enhance their own skills in these areas when they enter university.
Within such an environment the applied mathematicians, whether their special skills be in deterministic modelling, statistical modelling and analysis, numerical and computational techniques, or data storage and analysis, form their own professional group. A key aspect of their work is mathematical modelling  interpreting a new situation in quantitative terms, relating it to existing knowledge and techniques, and, to the extent of their own abilities and knowledge, developing appropriate procedures and software for handling the new situation. Moreover, through appropriate professional linkages, whether national or international, they are able to secure higher level advice when the problem moves outside the bounds of their own competence. Some professional mathematicians are employed as scientists within sector research organisations, some work out of universities or research institutes attached to universities, and some work within small professional companies or partnerships. A special characteristic of the professional mathematicians from a small country such as New Zealand is their flexibility, the speed with which they can bring their skills to bear on a new problem, even to a new field, and to collaborate effectively with colleagues with a range of different backgrounds.
In general, the fees that professional applied mathematicians can command in this society are commensurate with those in other professions. Their professional organisations support the free exchange of new mathematical ideas and techniques, but respect confidentiality requirements on data, and copyright for software products. (However there is an increasing tendency to make software available in the public domain, and increasing awareness of the benefits of doing so if the aim is efficiency and high quality in industry, manufacturing and business.)
Within largescale organisations, quantitative techniques have a wellestablished role, based on the gains in efficiency and quality that can be made by giving careful attention to the planning and design of the organisation's operations, to information flow with in the organisation, to objective, quantitative forecasting procedures based on the collection of appropriate data.
In the government sector in particular (but not only there), great emphasis is placed on collecting accurate and relevant data, on data integrity, on quality control methods in data acquisition, on visualisation and analysis of large data sets. The rapidly escalating scale of data sets, and the developments in automatic (eg scanning) methods of data acquisition, has led to the rise of a new "subprofession", combining elements of computer science, statistics, and numerical analysis, based around these tasks.
A particular difficulty for establishing a knowledgebased culture within a small society such as New Zealand has been found to be the small scale of many local enterprises. It clearly has been uneconomical for such enterprises to employ (as the larger organisations could) mathematical specialists on their own staff. On the other hand there were clearly national benefits to be gained from the improvements that quantitative methods could bring to efficiency and quality in smallerscale enterprises. To address this issue, a new type of professional consulting business has emerged (encouraged by some initial government support), whose staff could advise on aspects of information collection and flow in the organisation, the potential for savings and efficiencies through the use of modelling, statistical design and other operational research methods, appropriate software that could assist in such work, experiments that could help to isolate existing problems in production or forward planning. Such firms can be contracted, in much the same way as accountants, to give annual or special purpose reports, or to work with the existing staff in undertaking specific statistical, modelling or forecasting projects.
While much of the applied mathematical research in such a society is derived from local problems, some is directed towards problems of more general interest. Some of the strongest groups are in areas where local and international interests combine, as in modelling of geothermal and climatic processes, and the estimation and control of fish populations. Other areas where local groups have gained international recognition include quantitative modelling of risks from plant, animal and human diseases, and from natural hazards such as earthquakes, floods and landslides; genetic modelling for improved plant products; operational research methods for small enterprises. National funding agencies support such research both as a component in programmes of recognised output areas, and in their own right where longerterm development of methodology is required. Within the universities, the applied mathematics and statistics staff (as also some of their pure mathematics and computing science colleagues) participate in such research groups both within and outside the university.
Scholarship in pure mathematics is prized for two principal reasons: because the subject contributes directly to the intellectual heritage of mankind as a whole, it continually provides new methods and concepts for use in applications. Pure mathematicians are the ultimate resource when the mathematical knowledge of the applied mathematician reaches its boundaries. In such regards, the universities in the knowledgebased society avoid the error for which Bacon chastised the universities of his time, that "they are all dedicated to professions and none left free to arts and sciences at large. For if men judge, that learning be referred to action, they judge well; but in this they fall into the error described in the ancient fable, in which the other parts of the body did suppose that the stomach had been idle, because it neither performed the office of motion, as the limbs do, nor of sense, as the head doth: but yet notwithstanding it is the stomach that digesteth and distributeth to all the rest. So if any man think philosophy and universality be idle studies, he doth not consider that all professions are from thence served and supplied. And this I take to be a great cause, that hath hindered the progression of learning, because these fundamental passages have been studied but in passage". The pure mathematicians of this society maintain strong links with the international mathematical community, ensuring a flow of new ideas into research and applications areas. Many of them work closely with scientists both inside and outside the universities, and often derive their own research activities from these links.
Each one of the features listed above brings its own specific benefits to the knowledgebased society. Increase in reasoning skills and appreciation of quantitative arguments in the general workforce facilitates innovation, improves decisionmaking, and raises output quality and efficiency in individual enterprises. Indepth advice by professional groups saves money in improved design and efficiency, gives greater international competitiveness, greater ability to contribute to and profit from new developments in management, technology, information handling, and so on. (Illustrations of the sorts of benefits one would look for, albeit based on present examples, are listed in the section Contributions of mathematics to government, business and industry of the Report.) The greatest benefits, however, come not from any one feature in isolation, but the mutual reinforcement each gives to the other. High quality in teaching and research is reinforced by the practical use of and high regard for the mathematical sciences outside the educational sector, and conversely, the use of the mathematical sciences outside the universities benefits from the effects of high quality teaching, leading to a general shift of attitudes and spirit.
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In this and the following section we review the role of mathematics in government and the private sector, starting, as a means of gaining perspective, with a quick precis of the past, then looking in more detail at current issues, and trying to gain an impression of directions for the future.
The early history of mathematics in New Zealand, especially its uses in government and industry, appears at present to lack any systematic documentation. Only a short sketch may be suggested here. For this purpose, the history of New Zealand may be divided provisionally into four stages: the pioneering stage, up to the turn of the century; the prestatistics phase, from 1900 to 1940; the war and postwar period, from 1940 to 1980; and the postcomputer stage, from 1980 to the present.
In the middle of the 19^{th} century, scientific contacts with Europe flourished. New Zealand was a prime testing ground for the speculations about evolution, geology, natural history which captured both the scientific and the public imagination of the time, especially in Britain. As Ian Axford emphasised in his recent address, the first scientists to work in New Zealand were men of considerable stature. However, they were not mathematicians, but chiefly natural scientists  botanists, zoologists and geologists  primarily concerned with charting the new world which New Zealand presented to 19^{th} century science. New Zealand's strong traditions in these fields continue to the present day.
It seems likely that the first mathematics graduates to come to New Zealand worked here as lawyers, having obtained joint degrees from some of the older universities where law and mathematics were grouped together, even in the same faculty. A few found their way into the universities as early mathematics appointments; concerning the existence of others one can only speculate.
Even before this time, however, important practical applications of mathematics had been made by the early surveyors and engineers. By the 1860's the provincial governments had embarked on the construction of roads, railways, bridges, public buildings, and water supplies, a programme which was carried forward with great vigour by Vogel's government in the 1870's. The next two decades, under the forceful if not aggressive leadership of Sir James Hector, saw the beginnings of the National Museum, the Geological Survey, the Dominion Laboratory, and (in the form of the New Zealand Institute) the Royal Society of New Zealand. At a more pedestrian level, the keeping of demographic and economic data had been started by the provincial governments, becoming synthesised into a national department of statistics in the first decade of the present century.
The opening up of New Zealand during the 19^{th} century took place at a time of unprecedented popular and government interest in science. From the time of Captain Cook's voyage to observe the transit of Mercury from Mercury Bay, a thread of scientific, rational endeavour has run through New Zealand history; it is part of the New Zealand heritage, and although mathematics is not its primary focus, it provides encouragement for mathematical initiatives.
Throughout this period, the primary uses of mathematics were still in technical calculations for engineers, surveyors, chemists and other scientists, and to a lesser extent in actuarial and economic applications. It also saw the beginnings of statistical applications, not as a branch of mathematics, but for research purposes in fields such as economics, psychology, and the social sciences. Throughout this period, it needs to be borne in mind that a good practical mastery of calculus and analytical geometry was required to carry out computations that would now be done almost completely by computers, using specialist software. The expanding use of the telephone and of electricity created further openings for mathematically trained engineers. Mathematical training was important also in economic and social planning, especially in the Treasury.
A new dimension was introduced with the formation of the DSIR, and its vigorous promotion by its first Secretary, Ernest Marsden. From its early preoccupation with geology and chemistry, government interest in science turned quickly to the agricultural sciences. By the middle of the 1930's, at least Marsden, and his assistant (and successor) F.R. Callaghan, had become aware of the potential importance of statistical methods for New Zealand's primary industries, especially agriculture. J.T. Campbell, a student of A.C. Aitken's who had returned to New Zealand in the early 1930's, and in 1935 had been appointed lecturer at Victoria University, was the only local expert, and he was accordingly approached to set up a Biometrics section within the DSIR. Campbell demurred, preferring to devote himself to the needs of the university teaching programme, although at a practical level he was actively involved in advising Hudson, Ward, and others on the analysis of crop yields, and in early breeding trials. Ian Dick was subsequently approached, but the initiative was overtaken by the start of World War 2.
This period saw a great expansion in the role of mathematics, led by methods initiated or developed during the war: operations research, mathematics of radar and radio, coding and communication theory, stochastic modelling, theory of games, econometrics, etc. A number of young New Zealand mathematicians, involved in these developments in the armed forces, were recruited after the war to form the nucleus of the Applied Mathematics Laboratory (later Division) of DSIR. Its activities embraced probability and statistics, operations research, differential equations modelling, and computational mathematics. From 1950 until its absorption into the Industrial Research CRI in 1992, it acted as a focus for the application of mathematical techniques to local New Zealand problems, offering collaboration or consulting free to other DSIR sections and government departments, and, at a modest charge, to private industry, and helping in the practical training of young graduates. Its main clients were in agriculture and related disciplines (design and analysis of field trials, analyses of soil fertility, estimation and analysis of crop yields, breeding programmes), meteorology and geophysics (timeseries modelling of weather patterns, geothermal studies, earthquake occurrence), industry (process modelling, scheduling, quality control), forestry, health etc. During the 1950's, particularly, under the imaginative leadership of Ian Dick and later Robin Williams, with local guest star Peter Whittle, and a supporting cast of some half dozen able and enthusiastic younger scientists, it produced work of international importance, as well as securing much of the initial mathematical underpinning for New Zealand's primary industries. Later, its functions expanded and diversified, and it perhaps lost a little of its early verve, and the vision which had enabled it to capture projects of national importance (geothermal modelling being one outstanding exception). Also, recruitment of topclass mathematical personnel became difficult in the 1960's and early 1970's, as universities throughout the Western World embarked on a programme of rapid expansion in which mathematicians, particularly those with applied skills, were in short supply; a number of AMD staff were indeed lost to this source.
Also in the 1950's, a biometrics group was set up in the Ministry of Agriculture. The Ministry of Works expanded greatly, still employing mainly engineers but occasionally mathematicians and other science graduates, as did the Post Office in its work in the development of telephone services. The Department of Statistics also expanded, introducing automatic computing machines and a research methodology group, while research and development groups were started in Treasury, Health, Education, and most of the other large government departments. Many of the recruits to these were graduates with degrees in economics, but again some were mathematicians, and in most cases a background of at least first or second year mathematics was required. The expansion of the DSIR, and the development of the industry research associations, provided further employment opportunities for graduates in the mathematical sciences, or with a substantial mathematics component in their degrees. The New Zealand Statistical Association was formed in 1949 and the New Zealand Operational Research Society in 1963. Somewhat paradoxically, although underlining the utilitarian view of mathematics in New Zealand, the New Zealand Mathematical Society was not founded until 1974. A decade earlier, however, in 1966, as a result of an informal meeting of the university heads of departments, it had been decided to institute an annual meeting of mathematicians, which became the New Zealand Mathematics Colloquium, and quickly attracted interest from teachers at the schools as well as staff from the universities and mathematical research groups. Local Mathematics Associations, set up initially as links between local secondary school teachers and university staff, had been active even before that.
By the 1980's, a new phenomenon had taken root, destined to change greatly the shape of mathematics in government and industry, if not the substance of its applications. This was the advent of highspeed computing. Automation of the handling of financial transactions, automatic processing and analysis of survey and other statistical data, the introduction of digital methods in recording and transmitting of all kinds of signals and messages, not to mention the availability of powerful special purpose software for engineering, accounting, finance, design, statistical analysis etc., have transformed the role of the mathematical sciences. Far more mathematical computations, on a much wider range of topics, are carried out now than ever before.
The speed of computation has increased even more dramatically. Tasks that would have taken a team of computing staff a month to perform in the 1930's, or a day or so in the 1960's, are done in the space of a few seconds by nothing more than pressing a few buttons on a computer keyboard. At the same time new problems have arisen, particularly in the realms of capturing, checking, archiving, and analysing very large data sets, in optimisation problems with large numbers of variables, and in the increasing attention being given to nonlinear modelling.
Two other trends stand out in this period. One is the everincreasing diversity of uses of mathematics. Business and financial forecasting; the rise of the total quality management movement; new types of operations research problems; monitoring, analysis, and control of environmental variables; new telecommunication techniques; modelling and prediction of future growth of animal, fish, and insect populations; assessing the risk from the spread of infectious plant, animal and human diseases; and many others.
The other trend, somewhat paradoxically, is the reduction in the size of the mathematics groups within the government sector. This period saw the dismantling or substantial downsizing of most of the groups that had been set up in the previous period. One of the first to go was the Applied Mathematics Division of DSIR  downsized in 1986, then absorbed within the Industrial Research CRI, then further downsized. It was followed by substantial reductions in the size of the Biometrics Section in the Ministry of Agriculture and Fisheries, and of research and development groups in the other government departments. Some of the mathematical specialists made redundant in these exercises now act as consultants, either independently, or through specialist consulting groups in fields such as applied statistics, market research, database management, actuarial consulting, computergraphics, optimisation and linear programming, etc.
Overall, the changes which have taken place during this period are dramatic, ongoing, and not easy to interpret. The traditional careers for mathematical scientists and technicians have largely disappeared; the computations they performed can be done more quickly, more cheaply, and more accurately, by computers. On the other hand, the need to understand the mathematics on which the computer programmes are based has not changed  unless, at least, the organisation is willing to tolerate a high proportion of misdirected and inappropriate analyses. This is the challenge currently facing the mathematical professions: how to persuade managers that more than fast computations are needed to ensure the correct application and interpretation of mathematical techniques; and how to deliver the appropriate expertise.
The purpose of this subsection is to explore how mathematics is used in government, business and industry, and to indicate the financial benefits gained thereby. Systematic and reliable information on this theme is as hard to come by as it was for the previous subsection. The review team's questionnaires provide a starting point, but many aspects could be treated only partially. Further information of a nonquantitative character is contained in the Foresight Project, and in the extended reports set out in the Appendices.
When asked to estimate the extent to which the operation of their organisation was dependent on mathematical underpinning, the users responding to the questionnaire (see Appendix 2 for details) suggested figures varying from 0% to 100%. The gross figures indicated that, out of $1860 million total revenue reported by respondents, nearly $900 million could be described as being supported in this way. We take this as meaning that about 50% of the gross revenue is dependent on continuing mathematical support, in the sense that if the support were withdrawn, 50% of the revenue would be lost or put at serious risk. The costs of providing this support were estimated at around 12% (about one part in eight) of total costs.
The questionnaire returns are not enough in number to determine how the extent of mathematical support varies according to the size and character of the organisation. It can be stated with some confidence, however, that very few small businesses directly employ staff for mathematical support (at least leaving aside bookkeeping and financial accounting), or turn to mathematical consultants for advice and assistance. The above figures, therefore, give a rough picture of the situation for larger organisations, but would not be representative of smaller ones.
Within the organisations responding, a considerable range of mathematical techniques were reported. The most commonly quoted techniques were statistical (60% of respondents), followed by business mathematics and forecasting (50%). Process control and quality assurance methods were mentioned by 34%, and other operations research methods (including optimisation) by 22%. Engineering mathematics was mentioned by 31% of respondents, and physical modelling by 16%. More specialised topics mentioned included market research and surveys, computational mathematics, economic modelling and risk assessment. One user  perhaps not atypical  commented, "it is difficult to isolate our mathematical activities  they tend to permeate everything we do."
Respondents were asked to classify the staff undertaking mathematical work into four levels in regard to the mathematical tasks undertaken; of these, the first two levels are basic, the third requires a strong tertiary background in the mathematical sciences, use of degreelevel mathematics, and the fourth requires the development of new mathematical science techniques. Approximately 14% of staff undertaking mathematical work were listed as working at level 3. Only about 1% of staff are listed as coming into level 4.
About 50% of respondents indicated that they made use of outside suppliers for help in technical mathematical work  mostly in businessoriented statistical methods, such as market research, surveys, experimental design, and business forecasting, though some were seeking more specialised help in areas such as physical and process modelling.
To give a more concrete picture of the ways in which mathematical techniques had profited individual user organisations, respondents were asked to identify particular instances from their own experience. A number of these comments are reproduced below. Names of the responding organisations have been withheld to uphold the guarantee on the questionnaires that this information would remain confidential to the review secretariat.
Even greater sums, however, are involved in the control of animal or pest populations, and associated estimates of risk, which in turn depend on models for population growth, spread of infection, and other contingencies. For example:
Overall, there is an impression that mathematical techniques are not used sufficiently by New Zealand industries and manufacturers:
A separate questionnaire was sent out to research organisations  the CRI's plus a smaller number of independent organisations  covering much the same ground as that sent to the user organisations, but with some additional details. Again the details are given in Appendix 2.
As might be expected from scientific research organisations, use of mathematical techniques figures more prominently than in the user organisations. Out of a total revenue $280 million from the research organisations which returned questionnaires, nearly $190 million (67%) was estimated as being dependent on mathematical sciences. Overall, approximately 12% of the research organisations' budgeted resources were spent on mathematicsrelated activities, to underpin 67% of revenue. Four research organisations indicated that essentially all their resources were spent on mathematicallyrelated activities.
The relative frequency with which different mathematical activities were reported by the research organisations differed in some important respects from those reported by the user organisations. Statistics, quoted in 25 out of 28 responses, figured even more highly than for the user organisations, representing nearly 90% of responses, as against 60% for the users, perhaps reflecting the heavy concentration of the research organisations in primary products (14 out of 28 organisations, including most of the larger ones). In contrast to the users, the next largest group related to physics and process modelling, mentioned by nearly 50% of organisations, followed by computational mathematics and operations research, with about 20% each.
Considerable detail about the mathematical component in successful PGSF bids, which reflect mainly though not exclusively the bids from research organisations, is given in the earlier report by Menzies and McGlone. Suffice it to say here that, in all of the output sectors, there are major projects for which the mathematics component is of critical importance. The mathematical techniques involved cover a wide range, as indeed is evident from the information provided by the questionnaire returns.
A little surprisingly, the research organisations reported a smaller rather than a larger percent of their staff engaged on either level 3 or level 4 mathematical work (12% as against 15% for user organisations), although a larger percentage was working at level 4 (3% as against 1%). The research organisations were also more likely to contract out mathematical work to external suppliers (61% as against 46%). About the same percentage (33% of research organisations and 35% of user organisations) reported a need for expertise in the mathematical sciences not currently within the organisation.
Although it does not appear in the questionnaire returns, much of the mathematical work in the CRI's is undertaken by staff not initially trained as mathematicians. Indeed, there is some perception that mathematicians as such may not be of great value to the organisation. One CRI reports "I believe we are well served with the mathematics skills our staff in the physical sciences have acquired in the course of their education. I see little need for mathematicians, unless they also have training in physics or engineering. Biologists, on the other hand, generally have had poor mathematical training." Contrasting views are also expressed: "My experience is that (realworld) statistical issues/problems commonly require professional mathematician involvement, and are way beyond the typical engineer." What is common to both views is that, if mathematical staff are to pull their weight in a CRI, they need to be fully professionally competent, and able to put their mathematical training to practical effect in the research environment in which they find themselves.
This subsection summarises not only issues raised in the questionnaire returns, but also points raised in the MenziesMcGlone report, the foresight documents collected in Appendix 1 of this report, and the earlier LeipnikShen report. The issues are broadly grouped under subheadings below.
Many respondents have noted that substantial savings come from improvements in efficiency in large scale enterprises. The same percentage savings for a small business may not be worth pursuing, certainly not worth the employment of a mathematical specialist, possibly not even hiring a consultant. As one respondent noted, "Due to the size of NZ enterprises we are disadvantaged by the overhead required in employing highly qualified mathematicians". Nevertheless the advantages accumulated across a large number of small businesses could still be substantial in national terms. What initiatives can be taken to increase awareness of the possibilities, and to encourage smaller businesses to seek technical advice on mathematical questions? What kind of organisational structures would best service their needs?
A central issue, on which divergent views have been expressed, is whether specialist mathematical staff should be grouped together in a central resource, such as the old AMD, or should be placed as individual scientists within different research organisations or sections thereof. Proponents of the former view, mainly from the discipline, argue that a central group saves them from the problem of cultural isolation if they need colleagues with similar skills with whom to discuss procedures, clarify ideas, keep up to date with new methodological developments, etc. A respondent for one research organisation stated, "I'm sure there are (emerging areas), but I am not keeping abreast of these as I should." This is probably a common predicament for staff trying to work as mathematical specialists in isolation.
Leipnik in his report quotes US experience to the effect that the "critical mass" for a viable mathematics group is around 34 research "teams", with 710 members each. Without such a critical mass, problems arise in regard to library and computing resources, in coping with illness and staff movements, and in maintaining morale. Such recommendations may seem utopian from a New Zealand perspective, but in fact these are exactly the difficulties faced by the remaining mathematics groups in New Zealand user and research organisations, and they prejudice the delivery of toplevel mathematical services across the range of public and private sector organisations. The Applied Mathematics Division was the one group in New Zealand (outside the universities) able to provide such facilities, which is one reason why their demise has been so strongly felt. An overview of the current situation in regards to statistics is provided in the Table 1 from Bryan Manly's contribution to the Knowledge Base exercise, an updated version of which is reproduced below:
Table 1 Approximate numbers employed in Crown Research Institutes, research associations, government departments, and universities who are (or may be) engaged in statistical research for at least part of their time.
Organisation 
Researchers 
Totals 
Institute/Association 

Crop & Food Research Dairying Research Corporation Dairy Research Institute/NZ Dairy Board Forest Research Institute HortResearch Institute of Geological and Nuclear Science Industrial Research Limited Landcare Research National Institute of Water and Atmospheric Research 
14 
62 
Government Department 

Statistics New Zealand 
3 
28 
University  
Waikato Massey Victoria Canterbury Lincoln Otago 
20 
84 
Overall Total 
174 
A related issue is the apprenticeship period needed by young graduate mathematicians. Not surprisingly, both users and research organisations want staff with professional experience, who know how to face up to consulting or modelling problems in practical situations, and to work effectively with other scientists. But who is to provide young graduates with the initial experience? The usual procedure in a professional field is for new graduates to work for an initial training period as junior members in a larger organisation, but the only mathematical organisation left in New Zealand which works on a scale large enough to embrace this training function is Statistics New Zealand. The loss of the training roles formerly played by government departments such as the Ministry of Works and the Post Office has been cited as similar examples in relation to the training of engineers. As the groups are downsized, the training role becomes increasingly difficult to sustain, and no alternatives have so far appeared.
The extreme opposite viewpoint is that in fact New Zealand no longer needs any mathematics groups as such, but only scientists within specific organisations who have enough technical knowledge to apply the appropriate techniques or operate the appropriate software obtained from outside New Zealand. The scientists concerned do not need to be mathematicians, provided they have a sufficient knowledge of the relevant techniques. If it turns out that higher level mathematical input is needed, experts can be imported temporarily from overseas, or possibly from the universities.
Local professional groups would argue that such a policy altogether underestimated the difficulty of successfully using mathematical techniques in situations which inevitably involve some novel or nonstandard features. When mathematical problems arise it is not a question of just measuring the client's size and bringing out a jacket to fit. Even professionally trained mathematicians will find themselves up against problems outside their own range of expertise, but they will generally know whom to approach to assist them. Again the scale factor creates special difficulties in New Zealand, because the range of problems which can arise in a small country is no smaller than that from a large country, but the small country cannot afford to maintain the same range and number of specialists as the large one. Organisational questions become critical: how best can access to a wide range of expertise be provided with the limited resources available within a small country? Would specialists grouped into a small number of wellidentified national centres be more effective than the same number of individuals working in isolated situations? How does one best guard against the danger that, where the scale of the organisation does not warrant employing a fulltime mathematician, the mathematical components of the work "slip between the cracks"?
Less controversial is the view that within a larger environment, the ideal place for a mathematical scientist is as a full member of a research team, engaged as a scientist valued in his or her own right. Many mathematicians would agree. Certainly such a role places high demands on the quality of the mathematical scientist  perhaps more stringent than on those of the scientists in the major discipline, because more than just mathematical skills are involved  but those are just the mathematical scientists New Zealand would like to have. The questions at issue are whether mathematical scientists of suitable calibre can be attracted into such situations, and whether they can sustain their role in situations of relative isolation, without any further professional supporting structures.
A slightly different situation arises in some statistical work, for example in experimental or survey design, where at first sight the mathematical scientist is needed only for a rather limited technical role. However this impression is often deceptive. Commonly the application turns out to have some nonstandard aspects, and the best results are obtained when the statistician works closely with other team members throughout the research project. Once again the questions at issue concern the organisational structures that will best allow such links to flourish, and will attract and retain staff of suitable calibre.
A number of comments from both user and research organisations indicated concern about some aspects of new graduates recruited into their organisations. Some examples relating to high school graduates are as follows.
Other comments relate to university graduates in mathematics.
While universities can make note of such comments, they relate equally to the need for an apprenticeship period before new graduates can work effectively in a professional mathematical role.
A theme which recurs in comments from both government and private sectors is the financial cost of ignoring technical advice, of not seeking it in the first instance, of using wrong techniques, or using right techniques wrongly. Typical problems, which have seemed endemic in government departments and other large organisations in recent years, include the following:
Authors are understandably reluctant to be too specific in their criticisms, but the following examples are quoted in the questionnaire returns and individual submissions:
These concerns relate to two main points. The first relates to the output structures used as the basis of organising research bids. If the output structures (and indeed the CRI's themselves) are regarded as vertical, then mathematics, because it crosses structures, is horizontal. Skills are duplicated in different sectors, while bids for methodological developments are not funded, because they are not seen as the core business of any particular sector. Moreover, within a particular bid, the mathematical (especially statistical) elements are often regarded as peripheral, so that, under pressure, the mathematical components are often the first to be cut out. The result is a reduction in quality or efficiency in the output, coupled to problems for the continued employment of the mathematics staff. The net effect is a loss of capacity in the mathematical sciences across the whole public good science sector, and escalating problems in maintaining the quality of the mathematics component in current research.
The second point relates to a possible lack of quality control over the mathematics component in PGSF bids. Because of its supporting character, the mathematics component may not be looked at in detail, and the proposal may not be sent to reviewers with specific expertise in the mathematical aspects. As a result, technical points may be overlooked, resulting in inefficient design procedures, or inappropriate analyses. If there is loss of quality in the mathematics component of PGSF funded research, it may pass undetected.
One submission outlined a particular concern that where there are major issues in experimental design, data collection, data analysis, and overview analysis, at least one referee should have relevant data collection and analysis skills. The same submission expressed concerns about the need for overview studies of the needs for different directions of research, and lack of incentive for staff to supply them. Another concern is that it may not always be in the best interest of hardpressed science managers to seek out the most costeffective procedures. Provided the same outputs can be achieved, the larger the grant the better. What controls are in place to avoid possible abuses of this kind?
Most user and research organisations surveyed expressed some diffidence in their ability to envisage future trends in their uses of mathematical sciences, although increasing speed and sophistication of computer packages and associated data handling techniques was one general expectation. The following points have been obtained by putting together information from the questionnaire returns with the directions indicated in the contributions to the Foresight Exercise ( Appendix 1).
The first mathematician to be appointed to a university post in New Zealand was John Shand, MA (Aberdeen), appointed in 1870 to the Chair of Mathematics and Natural Philosophy at the University of Otago. The committee of the Otago Provincial Council which recommended the establishment of the University, required that "the branches of education taught by the different chairs should be of a thoroughly practical character, suited to the circumstances of the colony, and calculated to meet the requirements of the youths who will, in future years, take a more or less prominent part in its affairs". The four foundation chairs attracted 62 applicants each. Shand’s students praised his ability to clarify mathematical concepts. In 1886 he relinquished the mathematics part of his chair, but continued to serve the university until 1913, when he was close to the age of eighty. He acted for many years as Chairman of its Professorial Board and as Chairman of the Finance Committee of the University of New Zealand.
Despite the rigours of the pioneering environment, the distance from Europe, and the heavy teaching loads, the early mathematics professors included a number of considerable distinction. R.C. Maclaurin, the foundation professor of mathematics (also law) at Victoria University, who taught there from 18991907, went on to become President of M.I.T. from 19081920, and is highly regarded as a principal architect of the fortunes and reputation of that institution. Two geometers, D.M.Y. Somerville (Victoria 191534) and H.G. Forder (Auckland 1934 1955) both wrote texts of international reputation (Somerville's monograph on ndimensional geometry was still in print in the 1980's).
By and large, however, the principal work of the mathematics staff  initially just a professor, later more often a professor and an assistant, or lecturer  was in providing the basic mathematics (algebra, geometry, calculus) for students proceeding to careers in engineering, physical sciences, finance and banking, and secondary school teaching. It was not uncommon for the one or two staff to carry a full range of courses, from first year to fourth year, in both pure and applied mathematics, plus subsidiary (service) teaching, plus extramural teaching. Lecture hours were concentrated around the beginning, middle and end of the day, to accommodate the large numbers of parttime students. Whether they liked it or not, staff had little option but to follow the prescription laid down for Otago, that teaching should be "of a thoroughly practical character". If, under these circumstances, the universities’ ability to attract high quality staff should seem surprising, it must be borne in mind that until the second world war, the career opportunities for graduates in mathematics, even ones of excellent quality, were extremely limited. Apart from the universities  and university staffs were small  the options were limited to schoolteaching, a few jobs in actuarial mathematics or finance, and a few positions in industry and in statistics.
Research in the New Zealand universities was largely a postwar development. Before then, there was little time for it, and not much encouragement. Perhaps the principal service to research carried out by the early mathematics professors was in recognising talent in their students, and encouraging the best to proceed abroad, generally back to Britain, as in the link between Otago and Edinburgh, where A.C. Aitken supervised a number of New Zealand PhD students in the 30s and 40s. J.C. Beaglehole, writing in 1937, suggested that teaching up to that time had been done "without returning continuously to the laborious and galleried mine of research", to the end that in its universities New Zealand had "sought and ensured mediocrity with unusual success." There were exceptions, however, and Parton gives a pat on the back to T.H. Easterfield, Victoria's foundation professor of chemistry and later director of the Cawthron Institute, for instituting master's theses, supervision of which provided much of the stimulus for such research as did take place.
Lack of funds was the principal problem, and it was not until 1946, and as part of a more general reappraisal of the role of the universities, that the Chancellor of the University of New Zealand, David Smith, supported by C.E. Beeby in his role as Director of Education, was able to institute a regular programme for research funding. First through the Research Council, then through the University Grants Committee, the idea gradually won acceptance that research was part and parcel of an academic job, even in New Zealand. By the time of the Hawke Report in 1988, the universities could state as a general expectation that one third of the time of academic staff should be devoted to research and scholarship, with obvious implications on the teaching loads that academic staff could be asked to bear.
Even after 1946, research in the mathematics departments did not develop quickly. Continued teaching demands on small departments in a period of rising rolls, the ever increasing range of mathematical subjects and applications, the consciousness of older staff that they were out of touch with current research, the distance of New Zealand from major mathematical centres, and the relative difficulty of attending international meetings, all helped to inhibit the development of local research. Not until the 1960's, and even then only with considerable trepidation, did mathematics staff start to consider the possibility of accepting students for study at PhD level. Before then, virtually all mathematics students proceeding to advanced studies were obliged to travel overseas. Indeed, the differences in level meant that after arrival overseas, most such students were obliged to complete a further honours or masters degree before starting on their own research.
The 1960's and 1970's saw university enrolments in mathematics increase by leaps and bounds. Pure and applied mathematics (the latter meaning in the New Zealand context the application of mathematical techniques, especially differential equations, to problems in theoretical mechanics and continuum physics) had formed part of the traditional programme, but new demands arose for courses in statistics and discrete mathematics, and to supplement subject teaching in all faculties, especially in science and commerce. Table 2 gives an idea of the scale and speed of change. The heyday of the mathematical departments in the New Zealand universities in terms of enrolments in second year and higher level courses occurred during the late 1970's. At that time, the demand for engineers, physicists and chemists with traditional calculusbased skills was still strong, while demand for staff with skills in the areas of statistics, operations research, and econometrics, was rapidly growing. In addition, there was a chronic need for mathematics teachers to cater for a rapidly rising secondary school roll. After that time, the picture started to change again, with rising competition for students from computing science departments (often started within mathematics departments, but soon breaking away from them), and from quantitative areas of economics and finance.
Table 2 Staff in University Mathematics and Statistics Departments
Year 
No of Maths Staff 
No of Women 
1947 
10 
0 
During the 1970's, with intense competition internationally for qualified mathematics staff, the universities began to accept the need to build the mathematics departments up to a level where the teaching and research environment reached general international standards. This has been a major effort of the last two decades. With what success, may be judged from the material in Appendix 1 and the KnowledgeBase Exercise which preceded it. In his overview, Marston Conder points to several areas of outstanding strength, and also, as is inevitable in a small country, to some major gaps. Teaching resources, he suggests, still fall below international standards, as a result of general low funding levels. The problem of "critical mass", of developing research groups large enough to generate their own new ideas and enthusiasms, remains severe for all the universities save perhaps Auckland. On the other hand, communication via email, the reduced costs (in relative terms) of travel to international conferences, and the growing number of such conferences held within New Zealand or Australia, have facilitated the development of research in the mathematical sciences. Although research funding has improved  especially for conference attendance, for postgraduate students, and for postdoctoral fellows  it remains a significant problem. The advent of high speed computers has had just as dramatic an effect on mathematics within the universities as on mathematics outside them. It has changed the character of mathematical research, especially in the applied fields but not only there; it has opened new fields of mathematicsrelated research; and it has had profound effects on students' perceptions of the relevance and utility of mathematics. While computer applications have brought an awareness of the possibilities of mathematics to a far larger percentage of the population than ever before, they have not in general brought an accompanying awareness of their limitations or the factors required to use them effectively.
It will be clear from the comments at the end of the previous section that university mathematics departments in New Zealand are in a very challenged situation. They have made great efforts  commendable if not yet fully successful  to reach a position where they can participate in international research and offer facilities of international standard to both staff and students. This has been accomplished in a very short time, and during a period when the environment has been changing rapidly around them. Currently they are facing a period of university restructuring and possible retrenchment. Their support from the wider community is uncertain, and closely related to perceptions of the immediate practical usefulness of their teaching and research activities. At the same time they are aware of the pervasive underpinning role of mathematics, and the danger that the universities may once again find themselves "seeking mediocrity with unusual success". Just such factors lie behind the concern of university mathematics staff to review the role of mathematics, not only in the universities, but also in the community at large.
In this section we examine in greater detail some of the issues of more immediate concern to the university mathematics departments.
The rapid expansion of the mathematics departments during the 1960’s and 1970’s brought with it a problem of age distribution. It is revealed clearly enough in Table 3 below, reproduced from Appendix 2. Of the continuing academic staff in New Zealand university mathematics departments, over a half are aged over 45, and nearly a quarter are aged over 55. Less than 13% overall are women; in the 46 or older bracket, only 8 out of over 102 are women; and in the associateprofessor/professor bracket, only 1 out of 43 are women. Out of all mathematics and statistics staff, just 3 staff are listed as Maori, all males.
Table 3 Continuing Academic Staff by age and sex, Maths/Stats Depts only
Age 
Men 
Women 
Total 1997 
Total 2002 
<25 
1 
1 
2 
2 
2630 
3 
1 
4 
7.5 
3135 
13 
6 
19 
19.5 
3640 
24 
3 
27 
22 
4145 
18 
3 
21 
25 
4650 
28 
3 
31 
22 
5155 
29 
3 
32 
26.5 
5660 
24 
2 
26 
32.5 
>60 
13 
0 
13 
20.5 
Total 
153 
22 
175 
177.5 
There is an urgent need to try and redress the imbalance in ages, gender and ethnicity. The age distribution is of greatest immediate concern, because it affects also the vigour and choice of research fields within the departments. However, faced with a university system under duress, the possibility of recruiting younger staff, however excellent, appears unlikely, unless a special strategy is developed to address the issue.
Table 4, gives a picture of the growth in numbers of advanced students over the last two decades.
The disturbing feature is the lack of growth in student numbers, despite the increases in overall university enrolments (which have at least doubled over the same period), the increase in the numbers of mathematics staff, and the corresponding increase in the range of advanced topics available. In other words, the overall growth in numbers that has taken place in the quantitative sciences (and it is substantial if computer science and econometrics are included), has taken place outside the mathematics and statistics majors. Moreover the growth in staff numbers in the mathematical sciences, which until recently has largely kept pace with the numbers of effective full time (EFT) students, has not been fuelled by students seeking to study the mathematical sciences in depth, but by students requiring basic instruction through mainly first year service courses. This places the mathematics departments in a somewhat artificial position (albeit one matched by similar developments overseas), since the specialist skills of the staff are hardly being used to best advantage. Most recent growth, where it has occurred at all, has been in applied areas such as applied statistics, deterministic and stochastic modelling, operations research and financial mathematics, or in crossdisciplinary courses between mathematics and computer science.
Table 4 Number of Graduates in Mathematics, Statistics and Operations Research
Year 
Doctoral 
Masters 
B(Hons) 
Total Post Grad. 
Bachelor 

awarded 
Total 
% Fem. 
Total 
% Fem. 
Total 
% Fem. 
Total 
% Fem. 
Total 
% Fem. 
1974 
2 
0 
22 
4 
39 
36 
63 
24 
258 
24 
1991 

Mathematics and Computer Science Combined (separate figures not available) 

1994 
7 
28 
25 
24 
44 
16 
76 
20 
175 
43 
The net effect of these changes is to expose the university mathematics departments  and New Zealand's mathematics capability more generally  to a number of threats. The high average age level of staff affects flexibility and the coverage of new research areas. The likely retirement of a disproportionate number of mathematics staff within the next decade poses further problems. More advanced students are needed to take advantage of existing expertise, and more younger staff to help establish courses in developing areas. The present reliance on service courses to justify staff numbers is a potential threat to the stability of the mathematics departments, since enrolments can be critically affected by decisions taken outside the mathematics departments themselves, and over which they have little control.
Table 2 and Table 3 highlight the paucity of female staff in the mathematical sciences throughout the New Zealand universities, and Table A24 their nearabsence at senior levels. The signals to female students are clear: for whatever reasons, it is hard for women to get into mathematics careers, and harder still to get promotion. The proportions reflect the pattern in the 1960's and 1970's, when the bulk of current staff were recruited, more than they do the current situation; nevertheless it is still often the case that the majority of applicants for a vacancy in the mathematical sciences are male. This is also true of the very high quality graduates seeking employment in New Zealand from the Eastern European and Asian countries. Even a small number of women in key positions, particularly in chairs of mathematics, could alter the overall impression quite considerably, but it may not be easy to achieve such a situation.
Despite this somewhat discouraging example, female participation in most mathematics courses remains fairly stable at around 30% for pure and applied mathematics courses, and 40% for statistics courses (including statistics service courses). The same reference notes some differences in participation rates between institutions.
Participation by Maori and Pacific Island students is still low at both staff and student level. There is a handful only of current appointments who list themselves as being of Maori or Pacific Island origin. The proportion of students of Maori and Pacific Island origin in university mathematics courses is also small, perhaps 9.5%, compared to a proportion of 15% for university students overall, and of about 20% for the population at large. Common suggestions as to why these numbers should be so low include the low status attached to scientific and mathematical careers by Maori elders, and a cultural clash associated with traditional Western methods of teaching mathematics, particularly the emphasis on individual accomplishment.
By contrast to the Polynesian and Maori groups, the participation of Asian students in mathematics courses, particularly in graduate courses, far surpasses the participation to be expected on a population basis. It is quite common for onethird or more of students in graduate classes to be of Asian descent.
The reasons behind the student preferences quoted earlier are not hard to seek. Mathematics courses are difficult, they require dedication and special skills, and at the end of it all, apart from the few who proceed to research careers in mathematics, the career openings hardly warrant the effort taken  at least by comparison with students taking degrees in business, management, accountancy, computing science. Even that most traditional source of employment for mathematics graduates, secondary school mathematics teaching, has dwindled in attractiveness, as a result of modest salaries, and working conditions which are perceived as stressful and demanding. Only the combination of mathematics or statistics with other subjects  finance and management, economics, and computer science  looks tolerably attractive to a capable young person with an eye on a future career.
Some annual information about the destinations of new graduates is kept by individual departments, and information is systematically collated by the New Zealand ViceChancellors' Committee (NZVCC). The best recent source is the special study prepared in 1997 for the NZVCC and MoRST. In this study a questionnaire was distributed to all those who graduated with a degree or diploma from the New Zealand universities in 1990; it sought information on the character of their employment in both 1991 and 1996. Out of 12000 graduates canvassed, nearly one quarter completed returns. The resulting information gives (among many other factors) a good picture of relative salary rates in different professional fields. The following points are relevant to mathematics, including statistics and operations research.
For many years the mathematical sciences  including statistics and operations research  have been classified as an arts subject (MOE category A) for funding purposes, notwithstanding the considerable proportion of mathematics courses, at both specialist and elementary levels, that make extensive use of computing facilities. This low level of funding cramps the development of courses that would like to use such techniques more extensively, and pushes departments into a position where they are not able to match what is available in overseas universities. The mathematics departments have attempted to have their subject reclassified as MOE category B (semilab) but without success. While it can be argued that universities can internally adjust distributions from government grants and student fees, the lack of recognition of the actual fiscal requirements of these disciplines makes such redistribution extremely difficult to achieve. Until this or some other solution is found, mathematics departments (including fields such as statistics and operations research where the dependence on computer facilities is wellestablished at all teaching levels) will continue to be frustrated in their attempts to move their teaching in directions of manifest strategic importance, as well as immediate practical utility. The following comments from the responses to the university questionnaires indicate the general feeling on these points.
The concern about funding levels extends to funding for research. With the removal of "fundamental research" from the FRST output classes, the Marsden Fund is virtually the only external funding source now available for research in pure mathematics. However the success rate is too small to allow more than a small percentage of staff to benefit, at least directly. The applied mathematicians are in a better position to bid for FRST funding (and some of the more enterprising groups have done so with success), but they tend to meet the same difficulties as referred to in the previous section. (In fact recent changes to FRST funding criteria, particularly those encouraging cooperative bids, are likely to benefit mathematics groups, but it is too early to judge.) The net result is that all save a small proportion  perhaps 10%  of the university mathematics staff depend for their research support on funds from vote education, a dwindling commodity subject also to the uncertainties of university internal policies and restructurings. The situation is particularly difficult for young staff without the established track record needed to win Marsden Fund grants.
Again, computing figures as the single most important constraint. Table A34 in Appendix 2 shows total annual expenditure on computing hardware for mathematics departments of just over $750,000 for seven universities and a total scientific staff of nearly 300. This figure, modest though it is, does not fully illustrate the depth of the problem, for a significant part of the funding comes from research grants which in principle are at the disposal of the particular researchers obtaining the grant. In an extreme case, successful grantholders may have at their personal disposal more funds than are available for the rest of the department put together. Since only a small proportion of the staff are likely to win such grants, the outlook for younger staff attempting to work in computerintensive areas  now almost a necessity in virtually all fields of mathematics  is not encouraging.
Grumbles about the quality of preparation of incoming students are endemic among university mathematics staff, and no doubt common in other disciplines also. However there have been such major changes to both the school curriculum and the general structure of school education over the last several decades that some changes to students' preparation are to be anticipated. The following comments can be made.
Notwithstanding these difficulties, students' interest in the mathematical sciences remains high. Most recognise that they need mathematics for their further studies. The challenge is to match their needs with appropriate courses and teaching methods, and to encourage more to move on to higher levels.
The one computing science department which responded in depth to the university questionnaire made the following comment, quoted here in extenso.
"This survey has been very difficult for us to complete, as it is hard to categorise our staff with regard to activities related to the Mathematical Sciences. In the end we have stuck to staff whose main interests are in the area of theoretical computer science, although this has meant leaving out many staff whose research does involve a fair amount of mathematics. For example, we have omitted staff undertaking research in computer graphics, data compression, data communications, computer vision, image processing and artificial intelligence. We believe that this is a point which should be made in your report, namely, that there is a very strong link between Mathematics and Computer Science, with many crossovers where mathematical techniques are used in computer applications, and where computing techniques are very efficient at solving mathematical problems."
The relations between computing science and different fields of mathematics figures also in a number of the contributions to Appendix 1. Rob Goldblatt, reviewing developments in algebra, logic and foundations, notes that researchers in these fields are spread across both mathematics and computer science departments. The formation of the Auckland/Waikato Centre for Discrete Mathematics and Theoretical Computer Science is a practical expression of this linkage. At Victoria, the Mathematics and Computing Science departments recently combined to form a School of Mathematical and Computing Sciences, one aim of which is to foster closer links between the disciplines in both teaching and research. A more extensive structure including other related disciplines, such as the recently formed Institute of Information and Mathematical Sciences at Massey University's Albany campus is a development that should strengthen interdisciplinary links and lead to collaborative research opportunities.
If those developments reflect the algebraical end of the spectrum, comments of equal cogency can be made for mathematical modelling, statistics, and operations research. In statistics, for example, topics such as interrogation and analysis of large data bases, image processing, Monte Carlo and other simulation methods, are extremely close to computer science, but also have a crucially important element of statistical expertise. One concern is that with the shift of students, staff and funds into computing areas, applied projects will be initiated there which depend critically on such mathematical underpinning but are not supported by staff with the relevant mathematical expertise.
These changes reflect the changes which have so strongly affected developments outside the university. The extraordinary increase in the use of quantitative methods throughout the public and private sectors has been made possible by developments in computing. The computer application is what the customer sees, and is led thereby to suppose that the relevant expertise resides in computer science. The underpinning role of the mathematical component is lost sight of. These problems are at the hub of many current difficulties for the mathematical sciences, both within and outside the universities.
This is perceived as a problem for researchers in all the universities, perhaps a little less so in Auckland because of its size. As Douglas Bridges notes in regard to research in analysis outside of Auckland, "there are good analysts in the other universities, but they are essentially isolated, without daytoday close contact with a group of likeminded researchers." This is the classical problem for mathematical research in a small, potentially isolated university. Much the same could be said for most New Zealand academic staff in most fields of pure mathematics. The applied mathematicians, including statisticians, have opportunities for a somewhat wider range of contacts and in addition may engage in professional consulting, but similar remarks still hold in subfields such as probability theory, stochastic processes, biometrics, geostatistics, etc.
Improved electronic communication, and the encouragement to collective research brought about through the exigencies of the funding process, have already started to break down this pattern. As local staff begin to communicate with each other electronically, the opportunities for collaboration, and the benefits of closer contacts, are becoming more apparent, and encouraging the examination of further options. Several crossinstitute research projects are already in existence: the geothermal modelling project is one of the oldest and best established; the Auckland/Waikato Centre for DMTCS already mentioned is an example of a different type; the recently formed New Zealand Econometrics Study Group represents a third; recent FRST and Marsden grants linking Massey and Auckland, and Massey and Victoria, are examples of a fourth.
Such instances prompt a more radical reappraisal of traditional attitudes. Should the universities each continue to attempt to cover the full range of mathematical topics? Would it not be better for particular universities to establish their own spheres of expertise, in the knowledge that other universities covered other fields? The constraint is the need to provide a common undergraduate programme. This should not tax university mathematics staff too heavily, whatever their field of specialisation; indeed, there are already indications that the undergraduate programme has become too diversified, at the expense of a firm establishment of core knowledge and technique. At the graduate level the main impediments seem to be the lack of any tradition, such as exists in the UK, for students to move to another university for their graduate work, and of financial support for the students if they do so move.
Another aspect of this issue is the number of submissions and questionnaire returns which refer to the desirability for setting up research centres. A strong case is made by Graeme Wake and colleagues for a centre for mathematical modelling. Other suggestions include centres for nonlinear modelling, for official statistics, etc. Groups linked by electronic contact require no bureaucratic trappings: "the readiness is all". At the other end of the scale is a research institute with permanent staff and its own building, computing equipment etc. To be effective in joint research, a degree of direct personal interaction is probably essential. Consequently a centre with some physical location, with links to staff around the country who can be brought together for particular activities, and gaining some funding from research grants or commercial contracts, seems the type of centre most likely to add a new dimension to the existing situation
The Australian review phrased it thus:
"The principal output of academic research in the mathematical sciences is papers in learned journals and in conference proceedings; ..."
A Royal Society of New Zealand report recently analysed New Zealand authors' contributions to major journals. New Zealand authors produce on average over the period 0.43% of the total Mathematics papers worldwide (compared to 2.3% in Australia), compared with an average 0.49% of all papers over all science fields. As we show below, however, papers in Mathematics journals only account for slightly over half of output from New Zealand Mathematicians.
New Zealand mathematics papers are cited in journals at 87% of the rate worldwide (compared to 105% in Australia) and New Zealand receives an "Activity Index" of 0.82 in the Scientometric Indicator Datafiles which amounts to pretty much the same thing.
The cause is speculative: possibilities include the lack of research teams to produce "selfcitations", and the lack of people in "researchonly" positions who are far better equipped to keep pace with current research topics. New Zealand mathematicians are often forced to stakeout areas in which there is little interest worldwide. The lack of funding, and large distances involved in attending international conferences also makes "advertising" difficult.
A further point of interest in the Jasperse report is that the poorest performing field in the sciences is in Computer Science. If part of the reason for this is in deficiencies of equipment, this will also impact on computational mathematics (and increasingly on all mathematics).
Overall, one finds that mathematicians, when publishing in mathematics journals, are among neither the best nor the worst performing disciplines in New Zealand. However, this completely ignores mathematics' impact in other disciplines. The Australian review found that half of the papers produced by mathematicians appeared in nonmathematical journals, whereas (apart from informational sciences, with 17%), no other disciplines published in any meaningful numbers in mathematical journals.
As part of the survey, university mathematics departments, and departments with a strong mathematical interest, were asked to submit a list of their publications in refereed journals for the most recent twoyear period. Below we summarise the results, and compare them with the Australian experience.
Of the 175 fulltimeequivalent permanent staff in New Zealand, listed at that time as belonging to university mathematics departments, 104 (or 60%) have published at least five refereed papers in the last five years. This rate of a paper per year represents considerable effort in Mathematics for academics with heavy teaching responsibilities.
Dividing the journals into categories as in the Australian Review, we observe the pattern in Table 5.
Table 5 Percentage of articles by mathematicians in refereed journals, by field of journal
Fields of Research 
Australia 
New Zealand 
Mathematics 
48 
56 
From this we observe a few interesting points. Firstly, although a greater percentage of New Zealand articles are published in mathematical journals, there are few mathematical physicists working in New Zealand mathematics departments, and hence New Zealand mathematicians as a whole appear more applications oriented, and with more diverse interests, than in Australia. We must conclude, as in the Australian Review, that
In this way the mathematical scientists make a valuable, in many cases, essential contribution to the work of other disciplines.
Many trends and developments have been touched on in preceding subsections, and a more comprehensive coverage is provided by the individual reports in Appendix 1. Here we summarise some of the principal points. The aim has been to emphasise aspects which, for varying reasons, may be of particular relevance to New Zealand. Inevitably, there are overlaps between the areas selected, and there is a deliberate bias towards applications. The topics are not presented in any particular order.
This is an area in which New Zealand already has strength. Theoretical aspects include combinatorial theory, complexity theory, number theory, mathematical logic and programming structures. Apart from computer science there are important applications to operations research, mathematical genetics, etc.
The underlying motivation is understanding the behaviour of systems made up of many interacting components. Nonlinear features are typical. Early topics in this area included chaos and fractal theory, mathematics of phasechange, selforganising criticality. Both deterministic and stochastic aspects may be present. There are some individuals and small groups working in this field in New Zealand, both in mathematics and theoretical physics, but not enough considering that it is a fastgrowing area, already out of its first youth, with many applications.
The heading here is too broad, insofar as any application of mathematics to a real situation involves modelling of some kind. What is in view here more particularly is the application of mainly differential equation methods to the study of systems evolving in time. The two broad divisions are deterministic and stochastic modelling, but the two are coming together as more and more applications require the analysis of variability within the model and its forecasts. Applications of particular importance to New Zealand include geothermal and meteorological modelling, environmental modelling, economic modelling, modelling of biological populations, spread of diseases (epidemiology), growth of animals and plants, etc. There are some excellent groups already working in these fields in New Zealand, but mainly on the deterministic side; more expertise in stochastic modelling is needed.
Many problems in operations research and statistics involve largescale simulations, largescale optimisations, or both. Many technical problems arise in the application of even the best computer packages, and there is a shortage in New Zealand of experts with a good understanding of theoretical and technical aspects. The problems require a combination of computational science, probability theory, and operations research. Shortage of supercomputer facilities is a major obstacle.
This topic, colloquially referred to as "data mining", lies at the boundary of statistics and computer science. It has obvious relevance to the collection of data by government departments, but the application areas go well beyond that. Huge amounts of "transaction data" are generated by banks, supermarkets, and other businesses. Equally large data sets are generated by digital recording of geophysical, environmental and other measured variables. Questions of key importance involve the control of data quality, archiving and management of the data base, linking the data to effective graphical displays and relevant statistical analyses.
Financial mathematics (involving aspects of time series, probability modelling, and economic analysis), is a field which been the subject of major study and development overseas, but has been inhibited in New Zealand by the small scale of the local financial market, and the fact that many local financial organisations are in fact subsidiaries of companies with their head offices overseas. Risk management, taken in a broad sense, applies not only to financial portfolios but also to risks from natural hazards, and from the spread of agricultural pests and diseases. Local expertise in the mathematical aspects is very limited, whether inside or outside the universities.
The preceding sections also suggest a number of ways in which the activities of the mathematics departments may change or shift focus in the near future. Again we summarise the developments that appear most likely to bear fruit in the New Zealand context.
In almost all fields, New Zealand's university expertise is spread too thinly. To allow the development of research groups with adequate critical mass, concentration of expertise in a few centres is needed, with implications for the structure of teaching programmes at advanced levels, and the direction of graduate students.
Initiatives already undertaken are likely to expand, whether in the form of informal groups linked mainly by email, or specialised research centres linked to or operating out of university mathematics departments with encouragement and some more support funding.
Under the stimulus of rapid electronic communication, international as well as national research links are likely to expand, and to lead on to exchange visits, made possible by joint programmes funded from large national or international funding agencies.
This is an inevitable development, crucially important for bringing the teaching programmes up to international level. Sooner or later it will have to be recognised by improved funding arrangements, whether or not the mathematical sciences are reclassified by the Ministry of Education.
Links with computer science departments were discussed in depth in the previous subsection, and a movement in this direction seems inevitable. Quantitative groups in commerce and management areas, especially econometrics and operations research, to a lesser extent finance, management science, marketing and accounting, are facing similar pressures to the mathematics departments themselves, and stringencies in the universities may force them to join forces and work together more closely in designing and teaching suitable combined courses.
Unless other groups develop to fill the gaps, university groups in statistics and operations research, modelling, financial mathematics, and econometrics are likely to find themselves increasingly called upon to act as subject specialists in a consulting role for outside organisations.
The great bulk of students currently taking statistics courses receive only a very superficial understanding of both theoretical and practical principles. If the results of the questionnaires to users and research institutes mean anything, they mean a growing demand for statistical advice and assistance. Statistics enters into activities of organisations at many different levels: from grass roots up to the most sophisticated design and forecasting requirements. With far greater numbers using basic statistical routines through packages there comes a need for more expertise in the discriminating use of those packages. Departments should anticipate the need to increase the depth of understanding of statistical principles not only in courses for statistics specialists, but also for those studying statistics as an ancillary part of their degree programmes.
The character and quality of the mathematics programme in the schools has a major effect in shaping the general citizen's attitudes towards mathematics and how it can be used. The quality of the school programme is important also in attracting or discouraging students into the mathematical sciences, and in preparing them for further studies. Nor should it be forgotten that the schools provide an important source of employment for mathematics graduates. These are such crucial issues for the future of the mathematical sciences in New Zealand that it would be improper to conclude the review without some comments on them. The mathematics courses in the polytechnics are a further important component of the overall picture, directly relating to mathematical skills of the New Zealand workforce.
In fact, the school programme was not included in the review team's original brief, and although one set of questionnaires was sent out to mathematics departments in polytechnics (see Appendix 2 for a summary), the review team was not well constituted to examine their situation in depth. Moreover, issues relating to mathematics teaching in the schools are the subject of major reviews in their own right, sparked off in part by recent OECD findings on the performance of New Zealand children on an international scale. We shall therefore restrict our remarks to some summary comments, directed in particular to the impact of the programmes in the schools and polytechnics on the issues described in the earlier sections.
The most crucial stages in shaping attitudes are the earliest ones, so the family and the primary school are the areas most deserving of attention from this point of view. Just as positive attitudes towards mathematics can be greatly fostered by parent encouragement, so too negative attitudes, especially anxiety about whether the subject will prove too difficult, can easily pass down from parent to child, creating the elements of a vicious circle. To combat this danger, there is a corresponding need for a confident and vigorous mathematics programme in the primary schools.
New Zealand's participation in successive surveys of mathematics achievement coordinated by the International Association for the Evaluation of Educational Achievement (IEA) has led to a number of indepth studies of New Zealand children at different levels in the educational system. The second study (SIMS) was conducted in 1981 and focussed particularly on achievements of students at the end of their third form year; the third study (TIMSS) was conducted in 1994, and the two reports from this study released so far both focus on the primary schools. From the wealth of material in these reports we can pick out only a few key points.
There is no evidence of prevalent negative attitudes towards mathematics in the primary schools, particularly in the earlier stages. At standard 2 and 3 level nearly 50% of all students strongly agreed that they enjoy learning mathematics, with a further 30% or more who also agreed, if not so strongly. By forms 2 and 3, the number of strong agrees had dropped to around 25%, but still around 70% overall professed that they enjoyed learning mathematics. The numbers who found mathematics "boring" was around 25% in standards 2 and 3, but had increased to around 35% in form 2 and over 40% in form 3.
Many primary schools run extension classes in mathematics, and enter children for the mathematics competitions held in Australia (where the New Zealand children often do very well, and have sometimes topped the competitions).
Notwithstanding these results, New Zealand's overall performance is rather disappointing by international standards. This first came to light in the 1981 survey, when New Zealand was around the lower quartile of the countries reviewed. By 1994 New Zealand had moved closer to the median, but New Zealand third form students still performed weakly in questions related to decimal fractions, measurement, elementary algebra, all of which are fundamental for further mathematical study. Similar low performances are noted also at the younger age level, for example in relation to handling large whole numbers. The authors point out that some of these items are studied at a later stage in the curriculum in New Zealand than in many other countries participating in the survey; nevertheless the fact that New Zealand students tended to perform worse on average than their Australian counterparts is disappointing. It is also noted that the weakest results tended to be in questions relating to routine operations; New Zealand students often do quite well in questions requiring more initiative or imagination.
Commenting on the results of the 1981 survey, Clark and VereJones raise concerns about the lack of formal mathematics training among primary school teachers. They remark that, "In our discussions over mathematics education we were commonly faced with genuine surprise that anybody should consider mathematical knowledge beyond fifth form level at all relevant to primary school teaching". There were hints at that time that the selection of teacher trainees might discriminate against applicants with a stronger mathematical background, especially if they lacked other qualities considered relevant to primary teaching. Of course the knowledge is not required immediately for teaching purposes, but to give perspective and depth of understanding to the teaching of topics in the primary curriculum. Young children can ask remarkably penetrating questions about the basis of mathematical procedures, and it is important for the future attitudes, both of themselves and their classmates, that these are answered competently and confidently. The situation has certainly improved since that time, but the recent survey, while indicating that even at standard two and three level over 25% of teachers had some form of university degree, gave no information concerning the level of their mathematical background, so that the situation here remains unclear.
Overall, it appears that the mathematics programme motivates the children well while it remains at an informal, conceptual level, but that both teachers and students tend to lose confidence as formal elements (such as symbolic representation or manipulation) start to play a larger role. Gender and ethnic differences also play some role, and are discussed in detail in the reports quoted earlier.
There is concern among teachers that the curriculum is overfull, and that the constant assessment now a feature of the primary programmes is biting too deeply into teaching time.
The last two decades have not been easy ones for mathematics in the secondary schools. Continuing teacher shortages, delays in much needed rationalisation of old syllabuses, particularly in the school certificate area, problems in the changeover to unit standards, controversies about the appropriateness of such changes, uncertainty about the future of the examinations, increases in administrative loads connected with accountability procedures, and increasing behavioural problems among students, have all added to teacher load and teacher stress. Notwithstanding these difficulties there have been some notable achievements, in developing courses and procedures to cope with the wide range of students entering the upper secondary schools, in developing practical work and links with business and industry, in adapting the teaching programmes to a modular structure, etc. Here we have space to treat only a few key issues in somewhat greater detail.
The effects of such shortages have two immediate consequences, both of which tend to bear more heavily on students in their first year than at later stages. On the one hand, shortterm, ad hoc arrangements may have to be made for handling particular classes, amalgamating them with other classes, bringing in shortterm teachers, disrupting the establishment of any teaching "rhythm". On the other hand, teachers not trained in the area, or with weak mathematical backgrounds, may be asked to take the classes, establishing a precedent which may then continue for some years.
In the AucklandWaikato region in particular, where teacher shortages have been at their most acute, a new type of problem has arisen from the policy of recruiting overseas teachers to fill urgent gaps. Such teachers may be unfamiliar with the New Zealand education system, curriculum and customs. Extreme instances have been cited where not one of the mathematics staff is a native English speaker, and all have accents the students have difficulty in understanding.
At the time of the CERTECH report, overall participation in the final secondary year was extremely low by general OECD standards (in 1981, under 20% participation, as compared with over 80% for USA and Japan  but still comparable with England!). Various factors have been adduced for this situation; one of them was certainly the formidable examination structure (3 if not 4 external exams, with the key exam in Form 6 and not Form 7) in the last three years at school. During the last decade, the participation rate for the final year has increased from below 20% to nearly 60%. Inevitably, this has been accompanied by increases in the number of students taking one or other (or both) of the two mathematics course available (Mathematics with Calculus and Mathematics with Statistics). For example, the number of students offering Mathematics with Calculus for Bursary increased from 5517 in 1986 to 9211 in 1994 and then dropped slightly to 8772 in 1995. The number taking Mathematics with Statistics was even larger: 12107 in 1995, making it the single largest subject at bursary level (as it has been for a decade). Altogether 56% of the 1995 entrants took at least one mathematics paper.
These are substantial achievements, even if they do not put New Zealand at the top of the league table. In some fields, for example the practical component of the statistics programme, New Zealand is regarded as an international leader. However the figures still leave room for some concerns. In the last couple of years, the proportion of Form 7 students continuing with mathematics (though not the proportion of the age cohort) has dropped. In other words, of the broader spread of students relatively fewer are persisting with mathematics. Growing appreciation of the wider range of courses now on offer may be one reason behind this downturn. At the same time, there is concern in some quarters that loss of material on traditional mathematical techniques, and the need to cater for such a wide crosssection of students within the bursary classes, has depressed the standards achieved by the more able students.
Female enrolments for the bursary examination in 1995 were above male enrolments overall, but not in mathematics. About 42% of enrolments for Mathematics with Calculus were female and about 47% of those for Mathematics with Statistics; the differences come mainly from students taking both mathematics courses; slightly more females than males took one course only. Overall, Maori and Pacific Island enrolments remained a good 20% lower than would be expected on a population basis.
In such circumstances, the conditions of employment become of key importance in retaining old staff and attracting new ones. Stress arising from too large classes, obstreperous and uncooperative students, increased administration, lack of technical support, problems with education restructuring, etc, can tip the balance in favour of moving elsewhere, particularly if the alternative position is equally or more highly paid, and even if it is not. In all of these respects, the conditions of service for mathematics teachers have generally declined over the last decade, together with those of other teachers, but the mathematics staff tend to have better alternatives available.
Focussing on current difficulties should not allow the considerable achievements of the last decade to be overlooked. Since the relaxation of the examination regime in the mid1980's, the upper secondary school has been transformed, with far greater numbers of students attending, and taking advantage of a wide range of mathematics programmes. In addition to the regular programme, a lot of opportunities are now available to students through different kinds of extension programmes and competitions: the Mathswell activities, the Westpac competition, and the problemsolving competitions run by Otago University.
The polytechnics, much more than the universities, reflect the immediate reality of changing employment prospects; the quality of a degree may be as important as its content in determining employment prospects for a university graduate; the polytechnic graduate is more dependent on the immediate relevance of his or her training. For example, the change in demand for science and other technicians has seen a decline in many of the New Zealand Certificate programmes developed during the 1960's or earlier. In particular the New Zealand Certificate in Science is being phased out, while the New Zealand Certificate in Statistics has disappeared. The New Zealand Certificate in Engineering, numerically always the largest, is still continuing, though with reduced numbers. Its mathematics papers take students though a programme of practical mathematics somewhat commensurate in coverage with the applied components of a first and second year university mathematics programme, but with less theoretical underpinning.
What is likely to replace the certificates is not fully clear at the present time. A National Diploma in Science, based on credits towards Framework Units, is being phased in as the New Zealand Certificate in Science is phased out, and will have some compulsory mathematics and statistics credits. Plans for a National Diploma in Engineering are well advanced, and include Framework Units covering the old NZ Certificate papers 3031 and 4028 Mathematics, 4221 and 5290 Mathematics for electrical engineering. Unit standards in statistics (approximately first year university level) have also been prepared.
Increasingly, however, the polytechnics and technical institutes are moving to set up their own degree programmes, individually or in conjunction with other institutions, which may prove to be the real replacements of the old Certificate courses. Bachelor of Applied Science degrees, with some mathematics and statistics papers, have been available for several years now, and similar degrees in Technician Engineering are under development, and will include appropriate mathematics options at all levels. Such courses are likely to place increased demands on mathematics staff in the polytechnics, including a requirement to be involved in research, with all its concomitants.
A good indicator of the level of participation in mathematicsrelated courses is given by the numbers of enrolments in the paper 3031 Mathematics, which covers material at somewhere between the Bursary Mathematics with Calculus paper, and first year university courses in calculus. Annual enrolments are shown in the table below.
Table 6 Annual enrolments in the paper 3031 Mathematics
Year 
Total National Enrolment 
1990 1991 1992 1993 1994 1995 1996 1997 (provisional) 
1115 
The downturn in enrolments has occurred during a time of rapid expansion of numbers in the upper secondary school, and in the polytechnic system overall.
In their other programmes, covering a wide range of vocational courses, including business and information technology programmes, the mathematics component has generally been small, and may also have been reduced over the last decade. Some elements of statistics, computational procedures, use of packages or spreadsheets, would be common ingredients in what remains. In contrast to the universities, where a large proportion of students find themselves taking at least one service course in mathematics or statistics, no acrosstheboard demand for such courses has so far been expressed in the polytechnics.
The low demand for mathematics courses is reflected in the relatively small numbers of staff teaching in this area. This is shown in Table A36, which suggests that there are between forty and fifty such staff employed overall in New Zealand (not all the polytechnics responded, but most of those not responding were small institutions unlikely to employ more than one or at most two mathematics specialist staff, perhaps none). This figure may be compared with the 200 or so staff in the university mathematics departments, but the contrast is really more substantial since the university figure does not include staff in engineering, theoretical computer science, economics and econometrics etc, whereas the polytechnic staff includes many whose teaching overlaps into or is centred in these areas. In very broad figures, the polytechnics have about 40% of the total enrolments of the universities, but only about 15% of the total number of university staff in mathematicsrelated areas.
In all such discussions, the huge range in the size and character of the institutions loosely classed together as polytechnics needs to be borne in mind. Except in the largest institutions, or those with welldeveloped business or industry links, there is little scope for staff to undertake their own research or consulting (see the responses to questions 58 in Appendix 2, A2.5 Responses from Polytechnics).
The difficulties facing mathematics staff in a typical polytechnic (neither the smallest nor the largest) are summarised by one respondent as follows.
"We are struggling with the following: (i) very poor students, inadequately prepared by high schools; (ii) ineffective, weak prescriptions, leading nowhere; (iii) a unit standard framework which does not promote mathematics in any satisfactory manner; (iv) budget constraints which do not allow us to use new technologies in any form; (v) constant student perception that mathematics and mathematical science is "secondary" to their career path; (vi) teaching loads which preclude research or consultancy work."
Notwithstanding such difficulties, the mathematics components in courses such as Business Computing can and do provide new and useful learning experiences for their students, using commonly available software such as Excel, or MINITAB for statistics courses. The problems of scale and limited resources make it hard for the polytechnics to advance much beyond these basic ingredients in their courses. At the lower levels (Diploma and Certificate courses) there is continuing pressure from the subject specialists to reduce or downgrade the mathematics contents.
Nowhere is the fundamental predicament of the mathematical sciences in New Zealand shown up more clearly than in the polytechnics. Is it really true that mathematics is "secondary" to students' career paths; that good computer software, not a welleducated and numerate graduate, is all that is needed to utilise mathematics in a New Zealand environment?
Back to Table of Contents
In this section we attempt to address the fundamental questions raised in this report. How can the mathematical sciences best contribute to New Zealand's future? What steps can be taken to help them make this contribution?
Because the mathematical sciences permeate the daytoday activities of a modern society, the questions are broader than just identifying research areas that might repay potential investment. We give a brief outline of the general issues, followed by a more detailed attempt to tackle the problems posed by the Terms of Reference and the Foresight Project.
As surveyed in chapters 3 and 4 of this report, the situation of the mathematical sciences in New Zealand at the present time is uneasy. Superficially, they might seem to be flourishing. Never before have larger numbers of the workforce made use of quantitative tools. Software packages have put mathematical techniques, sometimes of considerable complexity, within easy reach of many. In the universities, Mathematics Departments have never been larger, and include among their staff specialists who are world leaders in their fields.
Looked at in greater depth, the situation gives cause for concern. The depth of understanding of the mathematical tools being used is often low, and the work may be undertaken in situations where there are few with the requisite background to perceive misuse or wrong interpretations. The easy availability of software, coupled to the financial stringencies facing most research organisations, are among factors which have lead to fragmentation and downsizing of existing mathematics groups. Outside the universities, the only substantial grouping of professional mathematical staff remaining in New Zealand, in either the government or the private sector, is in Statistics New Zealand (runners up might be the Reserve Bank and Treasury). Elsewhere, research and development sections have been reduced to a handful of individuals. The statistics and OR groups within the CRI's have met a similar fate. Actuarial and financial mathematics groups in insurance and finance institutions are also small. Even in the universities, the highly specialised mathematics staff are supported somewhat uneasily by the teaching of large first year courses to nonmathematics students. Advanced courses in mathematics and its applications tend to be unpopular with students who perceive them as leading to at best limited and (relative to the difficulties of training) poorly paid careers.
The underlying mood of many mathematical scientists at the present time, particularly those working outside the universities, is one of frustration, that the potential benefits that their subject could bring to New Zealand, or even to the organisation in which they are working, are not properly recognised. In the long term, when reviewed over a period of decades, the financial benefits that can be produced by mathematical modelling can be substantial, far outweighing the costs of the mathematical work. The questionnaire returns alone justify this view as far as New Zealand is concerned, and there is ample supporting evidence from other countries. Yet there are repeated indications that within New Zealand, more so than in most OECD countries, mathematical techniques are underutilised, and undervalued, particularly at management and government levels.
It is unlikely that the mathematical sciences will be able to contribute to their full potential in New Zealand until there is some alteration in this climate of opinion. What then can be done to move them from their present situation to something closer to that envisaged in the scenario in section Mathematics in a knowledgebased society. Some initial suggestions for components of a national strategy for the mathematical sciences in New Zealand are set out below. They will be discussed in more detail in later subsections of this section.
While any single step can help to change attitudes, a national strategy should look at tackling this problem simultaneously in different contexts. Moreover the suggestions above are not radical innovations, but seek to take advantage of trends which are already developing, both within New Zealand and overseas.
Some trends in the evolution and use of the mathematical sciences are likely to continue, irrespective of any special initiatives which may be taken. Following the discussions of future trends in the earlier sections and in Appendix 1, it may be helpful to summarise some key points before looking at possible special initiatives.
The general underpinning role of the mathematical sciences is not expected to change, although relative emphases may change within it, and its overall extent may increase. As quantitative techniques become yet more widely available through computer software, and the problems to which they are addressed increase in scale and complexity, there is likely to be a greater, not a lesser, need for mathematical understanding. Increased pressure on resources, on environmental planning, on monitoring and controlling physical, industrial and social processes, is likely to increase the demand for high quality quantitative information, and appropriate methods for its analysis and prediction. The cost of making errors is likely to increase, and with it the premium on correctly modelling the underlying processes.
The last two decades have seen remarkable changes in the professional role of the mathematician. The huge advances that computing science has made in speeding up mathematical computations, and developing userfriendly software by which the computations may be initiated or brought to bear on specific problems and data sets, has dramatically lowered the demand for traditional mathematics graduates, particularly at the polytechnic and first degree level. The work of the "statistical comptometrist", for example, or of the engineer who carried out standard stress calculations by hand, can be done more quickly and more cheaply by a simple software package.
The immediate impact of these changes has been to reduce the demand for mathematics graduates, and possibly to produce an impression that all mathematical problems can be resolved by the use of appropriate software. In fact, however, the developments in computing software, and the increasing scale and complexity of the problems to which it is applied, are tending to increase the demands for staff with a good understanding of mathematical principles. The easier it becomes for the novice to use mathematical software, the greater the need for backup staff who can provide help and guidance in its use and interpretation. In this sense, the availability of professional expertise in the mathematical sciences appears likely to be even more important in the future than it has been in the past. While these remarks apply to the mathematical sciences in general, one area which stands out in this regard is what might broadly be described as applied statistics and operations research. This is backed up by the fact that nearly all of the user organisations and research institutes surveyed saw additional expertise in these fields as one their most obvious needs over the next 10 years. On the other hand there seems to be little current enthusiasm for retaining the statistical groups developed over the previous few decades in the larger government departments.
The most likely outcome of these somewhat contradictory currents may be that these fields in particular will become increasingly professional, in a sense similar to that in which accounting and engineering are professions. Although methodology in applied statistics and operations research (just as in accounting or engineering) is continuing to evolve, there is now a large range of techniques which are broadly fixed, but which require professional expertise for their proper implementation and interpretation. Moreover the techniques are pervasive; they enter in at all levels in the operation of a business, industry, or government organisation, not just at the developmental or research level. While some areas of statistics, such as official statistics, agricultural statistics, and medical and health statistics, will continue to require their own groupings, in other cases a consultancy approach may be more likely to succeed. Such a tendency would be in line with developments by the mathematical professions to develop their own professional certification procedures and codes of ethics.
Overall, the mathematician may come to have more of an advisory and facilitating role than the technical role of the past. Any developments of this kind will affect the role of the mathematical sciences within the university, tending to increase their standing, and also the need to retain a strong core in the central mathematical disciplines themselves.
As a tiny country in international terms, New Zealand cannot hope to establish a scale of mathematical research which would allow it to match the range of achievements of countries with populations one, two or three orders of magnitude larger than its own. Where significant contributions have been made to mathematical research by New Zealand mathematicians, it has often been in one or other of two ways. One way is the serendipitous arrival of a mathematician (locally born or otherwise) who takes up residence in New Zealand, produces some major results while he or she is here, and inspires a surrounding group of colleagues. Such work may have direct value to New Zealand in its own right, particularly when it is inspired by local problems, but its value is more likely to be indirect, in lifting morale, in providing role models for young and talented New Zealanders, and in giving New Zealand international credibility in the mathematical sciences. Not a great deal can be done to systematically foster such developments, although it is crucially important to maintain working environments of high enough quality to attract mathematicians of international calibre.
The other way is through applicationsoriented research directed towards locally relevant problems, in which the combination of Kiwi ingenuity, and the relatively easy exchange of ideas between scientists in different fields, allows novel and effective approaches to be developed to problems which have resisted attacks by more orthodox methods.
Either way, the fundamental research undertaken by New Zealand mathematicians is likely to represent only a tiny proportion of the research that will actually be needed and used in New Zealand. One thing crucial for New Zealand's future will be the ability of New Zealand's mathematical scientists to keep abreast of new developments in a wide range of fields. Unless there are professional staff who have the background and skills to provide advice on the acquisition and implementation of new techniques as they arise, and to modify and apply them to local problems, there is a real danger of New Zealand's dropping behind general international levels, not only in the mathematical sciences themselves, but in all the other research and technological developments that those sciences underpin.
Very much the same level of scholarship and professional knowledge is needed for such advisory work as for fundamental research. Indeed, the provision of professional advice at this level should go hand in hand with research, and is likely to be closely linked to it. A key problem in New Zealand at the present time is that there are no centres in the mathematical sciences with this type of function. Some part this work can be provided by university staff, but they are hampered in this role by their teaching and administrative duties, and the lack of any specific university responsibilities in this regard. The lack of such centres also raises problems for the provision of library and other reference resources for mathematical scientists outside the universities, for the "apprenticeship" of young mathematical graduates, and for the development of a "critical mass" in given research areas. The situation has reached the stage where real concern has been generated and quite strenuous efforts are now being made to remedy the situation. This concern lies behind proposals to set up new types of research centres, possibly linked to the universities, and was a significant factor in initiating this review, at least from the disciplines' viewpoint. The problem is to find the right structural and funding framework to allow such developments to proceed.
In terms of broad directions within the mathematical sciences, a few points stand out clearly. One virtually incontrovertible point is that an essential component of the mathematics research of the future will be computingpower. This means that high quality computing facilities are going to be essential for virtually all areas of mathematical research. At the international level, exchange of software may become at least as important as exchange of research papers. Expenses are involved in purchasing specialpurpose software as well as appropriate hardware and distributed systems  a further argument for some degree of consolidation of mathematical research staff. It means also that the universities  indeed all tertiary level institutes teaching mathematical subjects  will need the ability to train graduate students and staff in computerintensive methods, including those associated with large data sets.
Beyond this, two currents of applied work stand out as having continuing local relevance both now and in the future. The first is the work in applied statistics and operations research which we have already identified as having a substantial professional component. The second is concerned with mathematical modelling.
Mathematical modelling, in its broadest sense, is the art of abstracting from a real problem the logical, quantitative elements that allow the essence of the problem to be stated in mathematical terms. The problems can arise in any field  so we have economic modelling, ecological modelling, physical modelling  and can use any of a wide range of techniques  so we have deterministic modelling, stochastic modelling, linear modelling, differential equations modelling, etc. Modelling of a more or less standard kind is a key ingredient of operations research and applied statistics, but in terms of mathematical research we want to reserve its use for situations where no standard models currently exist, or where the level of complexity is so great that essentially new ideas are needed to fuse together a variety of hitherto unrelated parts.
New Zealand has no deficit of situations requiring mathematical modelling. On a countrybycountry basis, the number of such modelling problems is not proportional to the size of the country's population; New Zealand has as many as most other countries. Nor is there any indication that the number of such problems is decreasing  quite the reverse is the immediate impression that one gains. Moreover the problems that any one country faces are not identical to those faced by others  analogous, in many cases, but not the same, so that tailormade solutions cannot be directly imported in the form of a computer package or an overseas expert (however useful both may be). To be able even to hope to cope with a wide range of problems with very few research staff, the staff have to be skilfully deployed, perhaps concentrated at the point where the essential mathematical nature of the local problems can be identified, so that appeal can be made to overseas literature, experts, computer programs etc, or the novel features identified which have to be cracked by the home team.
It is this key area in the mathematical sciences which is likely to best repay support through government science funding. The modelling problems themselves can arise (as indeed the MenziesMcGlone report makes clear) in any of the socioeconomic sectors. Environmental and population modelling, modelling of climate and tectonic processes, modelling of industrial and financial processes, economic modelling  no socioeconomic sector or field of application is immune. On the other hand, the mathematical tools required to solve the problem depend, not on the sector, but on the logical nature of the problem. Economically, it is neither feasible nor sensible to reproduce inhouse groups of mathematical modellers in every one of the research institutes serving different output classes. Some form of rationalisation and concentration will be necessary. The right combination of versatility and specialisation is likely to be the key for the successful development of modelling work in New Zealand, and this needs to be recognised and supported by appropriate institutional and funding structures.
The crucial question for the Foresight Project in regard to the mathematical sciences is the extent to which specific capabilities in mathematics have to be fostered, whether to maintain New Zealand's competitive edge in locally relevant research, or to ensure the successful running of government and private sector organisations.
In the preceding subsections we have emphasised the following points:
In the subsections which follow, we shall make some tentative suggestions as to how these issues might be followed up.
The terms of reference for the Report referred explicitly to identifying "opportunities for socioeconomic sectors within New Zealand in mathematical developments and the supporting needs for mathematical services." This is not a task that the review team has been able to accomplish. To do so successfully, it would require, not the type of general overview that the review team has attempted in the present exercise, but indepth discussions with experts in the different CRI's and in the different sector areas. From such discussions, it might be possible to identify, within the general research programme for each sector, those research topics where new mathematical developments would be required, or where existing mathematical techniques could make a substantial contribution to the effectiveness and efficiency of the programme.
In fact the review team would commend to MoRST the possibility of following up the present report with a more technical study of just this kind. It is not aware of any previous such attempt. Some of the problems of the existing mathematical groups, as well as the former Applied Mathematics Division of the DSIR, may have their roots in the difficulties experienced by such groups in locating the real problems, or in obtaining support for the problems that they could identify but which were not given priority by the organisations they were endeavouring to assist. Such a study could provide a firmer basis for the development of specialist mathematical centres, and would allow a better assessment of any case put forward for rationalising the grouping and funding of mathematical scientists within the PGSF/CRI framework.
At the present stage, we can only refer in general terms to projects or research directions identified within the earlier sections and Appendix 1, which seem likely to combine both national importance and substantial mathematical components.
In the first instance, these would seem to include
This list is illustrative only. Even within a particular sector, drawing out a more realistic list, assessing the strategic importance and research difficulty of different topics, as well as the nature and scale of the mathematical contributions which might be required, could present a task of considerable magnitude and difficulty. However, from the point of view of strengthening the role of the mathematical sciences, or foreseeing the mathematical capabilities likely to be needed over the next one or two decades, such an exercise could have considerable benefits, and might deserve repeating on a regular basis to ensure both sides keep in touch with new developments.
Leaving aside research tasks such as were the subject of the preceding point, there is still a major scope for the application of mathematical ideas to government, business, industry and finance through more or less wellestablished procedures. We have already referred to continuing demand for applied statistics and operations research methods, and to the fact that it is in these areas that the professional role of the mathematical scientist is most likely to develop. The primary initiative for such developments must reside within the professional associations themselves. The task of promoting the importance and value of their services, of securing appropriate financial and status recognition, is not one that is likely to be done for them. At present there appears to be some resistance to the greater use of quantitative approaches, as well as a need to reassess the numbers of professionals required, the character of the work they do, and the way they are organised. However these are all aspects which the professions will largely need to address for themselves. Ultimately, it is the reality of the benefits that the professional groups can bring that will determine the success or otherwise of their efforts.
There is, however, one particular difficulty in a small country such as New Zealand, which resides in knowing how best to allow small businesses to benefit from the improvements in efficiency and organisation that operations research or similar techniques can bring. This is a real problem, not a problem of attitudes or perception.
Part of the problem lies in better communication of information about the availability of techniques and of professionally qualified staff who can help organisations to take advantage of those techniques. This again would seem to be primarily a task for the professional groups themselves, even at the level of providing listings in such basic sources as the yellow pages of the telephone directory. However there could also be a case for a governmentassisted project first to ascertain the scope for such benefits within small organisations, and then to consider and assist in making such organisations aware of the possibilities. The benefit to an individual organisation might not be great, but it might be sufficient to provide greater competitiveness, particularly if applied nationally. A project backed by MoRST would carry more weight than one backed only by the professional groups.
In this subsection we examine some steps which might be taken within the PGSF/CRI framework to increase the contributions from the mathematical sciences to the public good science programmes, and to the New Zealand economy more generally.
The steps fall into two groups. The first relates to developments in the mathematical sciences themselves, or in the takeup of the mathematical sciences within New Zealand industry and manufacturing sectors. The question here is the extent to which these might be assisted or supported by the Ministry, the Foundation, or the CRI's. Several such points here have been raised in the preceding sections and we note the most important here for reference.
The second group of points relates to perceived difficulties in PGSF operating procedures in relation to the mathematical sciences. Ultimately, as mentioned earlier, these have their roots in the conflict between the "horizontal" character of the mathematics, and the "vertical" structure (whatever its other merits) used by the Foundation. Reconciling this conflict is not an easy matter, and the review team's suggestions are correspondingly tentative. That there is a problem, however, the team is convinced, and, from its discussions, believes this view would be overwhelmingly backed by the mathematicians still working in the CRI's around New Zealand. Three suggestions for possible consideration are set out below.
The justification for this new output area would be the essential underpinning nature of the mathematical sciences both in PGSF research and more generally. By raising the profile of good methodology it would encourage improvements in bid quality across all output classes. It would also reflect the importance of the mathematical sciences in leading developments into an era largely based on information technology (indeed, information technology could well be included in the title). Given such an output class, one could imagine one or more new centres or other organisations coming into existence for the specific purpose of improving the methodological aspects of CRI and other research. It would therefore address one of the current problems, whereby CRI scientists, who recognise a potential for developing improved methodology, cannot get funding to support such work. It would facilitate similarly cooperation between CRI's and universities or other organisations where appropriate methodologists were available.
This question was raised in the earlier report by Menzies and McClone. There appears to be some difference in opinion about the need for such control, and indeed over the quality of the work in current bids, between the mathematical professional staff and the FRST Output Managers or CRI CEO's. In fact the proposal has two aims. The first is genuinely to achieve a better degree of control over the mathematical components, particularly in the statistical work. The second is to ensure that the mathematical contribution is given proper weight, and that it is not always the first component to be axed if budgets have to be cut back.
It is clear that any proposal which substantially increased the bureaucratic component of the bidding process would be unlikely to be received with enthusiasm, and in the present case could well be unworkable due to the shortage of experts to carry out the assessment. Thus an effective proposal would have to require minimal additional effort in the bid preparation and would need to be largely selfadministrable. For example, bids might be required merely to indicate the level and extent of mathematical input required, possibly on a fourpoint scale. In addition, the source of the expertise needed  inhouse or consulting staff  could be explicitly indicated in the bids, and confirmed in the reporting. Even such a simple step would introduce an element of accountability into the performance of the mathematics component, and would help to ensure it was given proper attention.
A number of the CRI statisticians have suggested that the best way of securing core competences in areas such as statistical design or mathematical modelling would be to include the costs of such services as part of overheads for all bids entered by that CRI. This already works well in some cases, particularly where there is a strong tradition in the organisation of referring statistical issues back to the professional statistical group for comment and assistance. Its proponents suggest that this approach encourages the organisation to make full use of its mathematical staff, with concomitant benefits to the quality of the research undertaken.
This view is not shared by all CRI mathematicians and may be specific to statistical research in the agricultural and related sectors. The counterargument is that statisticians should pay their way along with all other scientists through specific bids in which they are a member of the bidding team. It is certainly the case that in ideal circumstances, the statistician or other mathematical modeller is best included as a member of the research team, not an outside consultant. There are two main difficulties with this approach in practice. First, the statistician (assumed male for ease of expression) quickly finds himself either spread thinly over many bids, or attempting to develop bids himself on methodological topics which are very hard to get funded within the standard output classes. Second, if bids are not fully funded, the statistical components tend to be regarded as having lower priority and hence get dropped from the bid. Both difficulties act to reduce the mathematical input to the research and the mathematical capability of the research organisation.
The points considered here have the general intention of shifting the centre of gravity away from too great a preoccupation with mathematics courses and research as such, and towards a greater concern with the broader role of the mathematics departments within the university and the community at large. Historically, the departments in New Zealand universities have an excellent tradition of interaction with the local community, whether in the form of links with mathematics teachers, or assistance to scientists and other personnel using mathematical methods. During the last two decades, the emphasis has shifted into building up research groups that can hold their head up in the international mathematical community. This is still a very difficult task, in an age of extreme specialisation and in the face of much larger and better funded specialist groups overseas. Two strategic advantages which New Zealand possesses, in mathematics as in other scientific fields, are its small scale and its traditions, deriving from its pioneering past, of ingenuity and mutual assistance. These both act to facilitate joint projects and interdisciplinary activities. At the present time, with both academic and scientific communities under stress, and particular problems emerging in the uptake and utilisation of the mathematical sciences, there may be some need to encourage deliberately initiatives which take advantage of these features. In addition, departments need to take seriously the fact that they represent the only major concentrations of professional mathematical expertise left in New Zealand, and have an obligation to make their expertise and their facilities more widely available.
In fact most university departments are aware of these issues, and many of the points listed below are either already under discussion or in the process of being implemented. Nevertheless it seems important to list them explicitly and reiterate their importance.
Many individual mathematicians already have links with scientific staff in the research organisations. What is needed here is a greater willingness on the part of the universities to look at flexible teaching arrangements that will allow such staff to take part in joint research programmes. The possibility of circulating or interchanging staff with the research organisations, allowing both staff the opportunity of extending their field of experience, was one explicit recommendation received; others related to the creation of joint centres. Joint research bids with external organisations can be encouraged within the university and the department.
To give such recommendations practical teeth, some concessions will need to be made by the universities. The increase in the range of activities undertaken by the departments will require some support through additional facilities and staff. Even if, say, research bids from university staff fully cover their salaries, additional expenses are likely to be incurred in attracting and deploying replacement staff and providing them with facilities. The university may have no money it can spare for such developments, but it can at least assist indirectly in a willingness to accept ad hoc arrangements and flexibility in staff management.
No other development has had such a great impact on both courses and research in mathematics as the computer. Computers are now used routinely in course work, research, and consulting. The development of excellent mathematical and statistical computing facilities should be a goal of all departments. Without them it would certainly be impossible for departments to play successfully the wider professional role envisaged in this section. At present the achievement of such a goal is frustrated by an apparent unwillingness of university budget managers to recognise the key importance of computing in mathematical fields. The mathematics groups request that their subject be reclassified as a class B (semilab) subject, alongside Computer Science and Psychology. While this is one possible path to securing a sounder financial footing for the mathematical sciences, it could introduce some additional problems of its own. For example, it has consequences for student fees, which could be damaging, and it does not apply uniformly across all types of mathematical activity. What is essential, however, is that the university and its funding authorities recognise the great importance of providing proper computing support for students advancing in presentday mathematical (and not just computing) subjects, and for staff engaged in mathematical research.
Reference has already been made to the dependence of current mathematics departments on the socalled service courses offered to firstyear students in nonmathematical subjects. Relegation of such students to largescale, mass education programmes, from which it is very hard to gain any real appreciation of the force of mathematical processes in any real context, may itself be a contributing factor to the apparent low level of esteem in which mathematics appears to be held by current managers and politicians. All factors point to the need to take very seriously the responsibilities involved in such programmes, at two levels. The first is in regard to the quality of the service courses themselves, in particular their quality as seen by the departments whose students provide the EFTS to keep the mathematics departments running. Close collaboration with the "client" departments in selecting material and method of treatment should be encouraged. There is a strong case for considering also joint teaching arrangements, which can include special tutorial groups or shared lecturing or guest presentations. Ultimately the most secure arrangements will be those in which both the service and the client department are confident that the best combination of teaching responsibilities is in place, including if necessary some sharing of the EFTS benefits.
The second level is in catering for the continuing needs of students in other disciplines for mathematical support and advanced training. More effort needs to be made to shift teaching effort into such arrangements and away from narrowly specialised courses for a handful of students advancing in mathematics. Jointly taught second and third year courses, methods courses for postgraduate students, and more consideration for the needs of nonmathematics students in course planning and development are all factors worth revisiting. Many such courses will be computerintensive, since facility in using different types of mathematical or statistical packages, and understanding and discrimination in their uses, are often key concerns of such students.
What is in view here is not private consulting as a means of salary supplementation, which is accepted by most universities under some constraints, but consulting and contract research carried out through the department and seen, not only as a moneyspinning exercise, but as part and parcel of the university's service to the community. Nearly all those who have had direct experience of such activities, recognise that they come at a high cost in effort and stress, since consulting for external bodies has to meet constraints which are difficult to combine with the university's own requirements for deadlines in the teaching programme. At the same time they play a role equal to, if not stronger than, pure research in enhancing the staff member's ability to talk with conviction and experience about current uses and techniques. The need for greater involvement by mathematics departments in such activities stems from the earlier recognition of the shortage of other sources of highlevel expertise, and the need to make such expertise available outside the university. Ideally, the university department should be a recognised centre to which mathematicians and mathematics groups outside the university can turn for upskilling, retraining, research leave, and the initiation of joint projects. Consulting, contract research, and advanced professional courses run by or in conjunction with university staff, are all components in such a programme, and are activities which will be needed if the mathematical sciences are to flourish locally.
At a practical level, the difficulties facing such developments are of two kinds. The first is in persuading university administrations to put their weight behind facilitating such activities. Stringent overhead regimes, unwillingness to bear risk associated with any form of entrepreneurial activity, inappropriate and inaccurate financial reporting procedures, and inflexibility in developing novel types of combined teaching, research and consultant positions, are among the problems likely to be encountered. While it would be generally accepted that such activities must be fully financially selfsupporting, and even to make a modest contribution to general university revenue, there is a need to recognise that the greatest benefits are likely to come from allowing the specialist groups to plough back surplus income into upgrading computing and other support facilities, even including additional support staff.
The other practical difficulty is in finding academic staff willing and able to take part in these activities. They are not easy, and some positive inducements are needed to prevent staff from taking the easy options of remaining within a familiar teaching role, concentrating on their personal research programmes, and using such spare time and energy as they possess to increment their income through private consulting.
In times of constrained university funding, course rationalisation is one area where the mathematical sciences could play an important role. The duplication of course content, especially in papers involving research methodology, should be of concern.
A further extension of the teaching role of mathematics departments in is the possibility of joint appointments. There are obvious opportunities in areas such as mathematical physics, computer science or mathematical finance, which have a high mathematical component. However the increasing use of quantitative methods right across the spectrum of university disciplines opens up the possibilities for joint appointments in less traditional areas such as mathematics and psychology, mathematics and biology, or mathematics and linguistics. Such joint appointments can lead to, or be caused by, joint courses at the undergraduate or graduate level, and joint involvement in research projects. The main impediment is inflexible departmental and faculty structures within the university, which serve to perpetuate rigid or even possessive departmental and faculty boundaries. Many universities pay lipservice to the need for such joint activities, but few support them by providing appropriate promotion opportunities or structural and administrative backup.
Lest it be thought that these suggestions place too much emphasis on the professional and training aspects of mathematics departments, it may be desirable to reiterate the need for strength in basic research areas. The diversity of such areas is so large that it is barely possible to conceive how they could be covered by the mathematics staff of all universities in New Zealand combined, let alone in any single university. Some rationalisation and concentration of resources in one centre or another is likely to be necessary, and part of the price to be paid for having an effective programme overall. Such rationalisations aside, however, it should be clearly understood that the core and centre of a strong mathematics department are its own specialists, who make available to their colleagues, in their home university and elsewhere, as well as to their students, their scholarship and expertise.
Despite the difficulties referred to in the section Mathematics in the schools and polytechnics, the mathematics programme in New Zealand schools has often been innovative, and supported by dedicated and enthusiastic teachers. Outcomes such as the poor relative performance of New Zealand children in mathematics assessment tasks are the more disappointing from this point of view.
The two fundamental problems which seem to underlie them may be, firstly, the recruitment and retention of capable staff, and, secondly, the conflict between the contents of a mathematics programme that can be taught successfully to selected groups of students by able teachers, and one which can survive the rigours of a mass teaching programme in which children of all levels of ability and background are taught by teachers with a comparably wide range of talent, enthusiasm and experience.
It is outside the scope and competence of this report to make recommendations in either of these areas, but it is important to draw attention to the importance of the role of the schools in shaping the overall picture of the mathematical sciences in New Zealand. Schools are the place where initial attitudes are formed and skills learned. It is through their training in the school programme that students will develop the ability to apply sound logical principles to real problems, and to handle and interpret quantitative information. These are not skills which of themselves involve a great amount of technical mathematics, but they seem difficult to acquire without some technical training in mathematical subjects. Some technical background is needed also to interpret quantitative arguments and to apply specialist software.
Within the school programme, nothing is more important than the quality of the teachers. Good teachers can make a success out of indifferent programmes, but the converse is not true: even the most excellent programmes can be frustrated by teachers who do not understand their purpose or content. The strategic importance of the recruitment of high quality mathematics teachers should be emphasised in communications between the Ministries, and any steps thought necessary to achieve this recruitment, in terms of financial recognition or professional support, should be given careful consideration.
The task of devising and implementing a curriculum that can be applied to schools catering for students with the most enormous diversity of interests, skills, and backgrounds, is a formidable one indeed. The best approaches to this problem, which is by no means New Zealand's sole prerogative, are still being worked on. New school programmes in mathematics have recently been introduced, giving, inter alia, greater importance to problemsolving skills. Their impact needs time to be assessed. They are only one of many innovations to have affected the schools in the last decade. There is an urgent need for a period of stability to assimilate these changes, to sift the wheat from the chaff, and to shift the emphasis back from the management of restructuring to the teaching of students.
Any steps which can be taken, from the primary schools right through to the seventh form and even into the polytechnics, to boost the mathematics programme in the schools, or to encourage children to develop their logical and quantitative skills, will at the same time assist New Zealand on its path to the future.
Last modified 14 July 1998. Final Version.
Edith.Hodgen@vuw.ac.nz