The Mathematical Components of Engineering Expertise: Project proposal (Submitted to the ESRC in April 2000) Professor Richard Noss & Dr Phillip Kent Mathematical Sciences Group Institute of Education London WC1H 0AL Funded by the Economic and Social Research Council (grant number R000223420), February – December 2001 http://www.ioe.ac.uk/rnoss/MCEE SUMMARY The project will examine how mathematical ideas and techniques are used in the practice of engineering by undertaking detailed observations of professional civil engineers working on design projects in a large civil engineering firm. The proposed work builds on a previous ESRC-funded study (the TMO project) which investigated the mathematical practices involved in the work of nurses, bank employees and commercial pilots. The most basic finding of this study was that the mathematical elements of workplace settings are subsumed into routines and tools, and do not play any autonomous role within them. However, a more careful analysis revealed another face of mathematics intertwined with professional expertise, where judgements are based on models of situations, having both qualitative and quantitative aspects. This new project seeks to extend this work by investigating engineering practice, a workplace setting in which the role of mathematics is more explicit and sophisticated, and where there is much greater recognition on the part of practitioners of the roles that mathematics might play in their work. Since we will be studying what actually happens (rather than restricting ourselves to asking people to describe what they think they do) we hope to understand better the mathematical elements of the routine, implicit and unquestioned aspects of engineering practice. This will allow us to draw conclusions on the kinds of mathematical knowledge which are actually employed, and how this knowledge connects with the decisions and judgements that are made by engineers in their work. It will also allow us to draw useful conclusions concerning the kinds of mathematical education which are appropriate to engineering students, a topic which is currently the subject of much debate within academic and professional engineering circles.

1

BACKGROUND TO THE PROJECT In an earlier ESRC-funded study (1995-98), “Towards a Mathematical Orientation through Computational Modelling” (TMO), we studied the mathematical practices involved in the work of three groups of professionals—nurses, bankers and commercial pilots. In this new study, we want to build upon the results of TMO by looking at a group of professionals whose work involves a more complex repertoire of mathematical ideas, and we will take the specific example of civil engineers, observed at work in a large engineering consulting firm. The workplaces and the practitioners studied in TMO stand in contrast to those analysed in most previous studies of workplace mathematics, where researchers have sought to relate general kinds of “school” mathematics to “typical” everyday working practices—for example Scribner’s (1986) study of the dairy industry; Lave’s (1988) investigation of weight-watchers, and the work of Nunes et al (1993) on street vendors. In contrast, the TMO studies involved three professional groups which are similar in that they all have well-defined mathematical entry requirements developed through specific training programmes. These programmes usually try to provide students with somewhat general mathematical techniques that they will be able to apply in practice; yet one of the most notable observations in the study was the way in which these general techniques were not used in many elements of the practices that we observed, being instead replaced by a set of strategies dependent upon context. The TMO study sought to examine the epistemological and psychological issues involved in relating mathematical and professional knowledge. Moreover, the intention was to define mathematical knowledge in its broadest sense, and to include any activities which involved the mathematisation of workplace activity, not just those that consisted of fairly direct application of taught mathematical techniques. This allowed the study to gain a better understanding of how those essential mathematical elements focussed on in training were put to use in the realities of practice—and what other “mathematical” (in the sense of abstracted or generalised) understandings were evident. As a result, the TMO research has led to several detailed analyses which have shown that the various professional groups employed a variety of mathematical strategies which were finely tuned to their practice, yet simultaneously retained the notion of invariance that typifies abstract knowledge. For example, in the case of nurses, the invariant of “drug concentration” was found to underpin a set of proportional reasoning strategies to calculate drug dosages, tied to specific quantities and volumes of drugs, the way drugs are packaged and the organisation of clinical work (Hoyles, Noss & Pozzi, in press). From a theoretical point of view, these findings involve abstraction within a specific setting. Although abstraction has been a major concern of mathematics education research since the 1980s, most notably in the “advanced mathematical thinking” (AMT) tradition, research has focussed mostly on the learning of “pure” mathematics in university and post-compulsory schooling (see, for example, Tall 1991, 1999; Czarnocha et al 1999). In the past, we have criticised AMT’s view of the growth of mathematical understanding through a sequence of specific stages, and we have proposed (Noss & Hoyles, 1996) an alternative theoretical framework for abstraction in terms of •

webbing — a description of the ways in which learners come to construct new mathematical knowledge by forging and reforging internal (mental) connections through the interaction of internal and external resources during activity, and

•

situated abstraction — that learners construct mathematical ideas by drawing on the webbing of a particular setting which, in turn, shapes the way the ideas are expressed; that is, abstracting within, not away from, situations. 2

We intend this framework to be applicable in traditional mathematical learning situations, and recently we have looked in detail at how mathematical knowledge becomes abstracted in the contexts of 10-11 year olds’ construction of meanings for randomness (Pratt & Noss, submitted), and undergraduate-level applied mathematics (Kent, 1999). But we also intend that the framework should apply to situations where mathematics is being applied to other knowledge domains, and so, in the proposed study, we now want to extend our efforts further in the direction of situated abstractions of mathematics in professional work. RATIONALE OF THE PROJECT To build upon the results of the TMO study, and to continue to develop a theory of abstraction, we need to move the research into a domain of practice where we can see more sophisticated kinds of mathematics (geometry, algebra, calculus) that entail more complex structures of abstractions in their practical application. That is, we would like to investigate the problem of characterising mathematics in professional practices where mathematics plays a more overt, ubiquitous and complex role. In our chosen domain of civil engineering, we will be examining the process of engineering design, that is, devising a plan to carry out an engineering project before doing it. A design plan is typically very complex, and created over a long period of time—months or even years. Mathematical calculations, although crucial, form only one part of a set of results that support the final plan. Moreover, engineering design is done by teams of people (ranging in size from a handful to a hundred or more), made up of different kinds of specialists, where mathematical work may be the principle responsibility of only a small part of the team. We have a particular interest in the role of digital technologies in this collective design process: is mathematics becoming an ever more specialist realm, where non-specialists are increasingly just the consumers of “pre-cooked” mathematics hidden behind the interfaces of software packages? We believe that this question is not as clear cut as it seems, if we admit (as we did in TMO) a broader definition of mathematisation than the visible application of mathematical techniques. In the TMO study, we regarded the abstractions of mathematics into the professional practice as essentially single-layered: for example, a mathematical concept, such as ratio, is embedded into the practice of drug administration. We see engineering as a broad and important example of a more complex, mathematically-rich domain, where mathematical meanings and understandings are shaped, epistemologically and psychologically, by the activities of professional practice. Moreover, as Hall (1999) points out in the context of architectural design practice, mathematical meanings are socially negotiated in workplace settings, with multiple interests and different accountabilities driving the character of mathematical description within a given project. We intend “mathematically rich” in the sense not only of more mathematics, but also in the multi-layered structure of abstractions, in which a fragment of mathematical knowledge becomes embedded within another piece of mathematical, or scientific, or engineering knowledge—perhaps recursively. We propose to study how this multi-layered structure of engineering and mathematical knowledge becomes embedded into engineering practice. Relatedly, while there is no doubt that engineers use a great variety of mathematical concepts and techniques, this does not necessarily mean that it is mathematics in the (mathematician’s) sense of generalised, abstract knowledge. For example, in Kent & Noss (2000) we made a comparison of different mathematical software packages, some of which are marketed with a target audience of mathematicians and theoretical scientists, whilst others have a target audience of engineers. We think it is quite 3

intentional on the part of the software designers that the former type of package aims to present itself as a mirror to the mathematics knowledge domain, whilst the latter type treats mathematics as a “toolbox” of techniques for use in “applied” (engineering or scientific) problem solving. This comparison of perspectives suggests to us that mathematical epistemology is not a simple issue: it depends on what software designers, and users, think “mathematics” is. The answers to such epistemological questions have far from esoteric implications. The “application” metaphor underpins the mathematical training of undergraduate engineering students in the UK: in service mathematics courses, they are taught “the mathematics” that they will afterwards learn to apply to their engineering subject. There are clear signs that this curriculum system is coming under considerable strain: for example, two reports commissioned by the UK engineering institutions (IMA, 1999; Sutherland & Pozzi, 1995), have highlighted the problems caused by the significant changes to the teaching of mathematics in schools since the 1980s, with many students now lacking the kind of mathematical foundations which traditional service mathematics courses demand. We believe that our study of engineering practice will provide valuable information about the adequacy of the application metaphor as a basis for understanding how professional engineers use, and learn, mathematics, and that our findings will be of use to teachers of undergraduate engineers in developing revisions to service mathematics curricula. METHODOLOGY AND TIMETABLE We propose to take a broadly ethnographic approach which aims to analyse the material and social organisation of knowledge-in-use within a specific design project in a single large civil engineering firm. The early part of the study will involve gathering initial information, and negotiating with the firm the design phase of an engineering project for close and detailed observation. The criteria for choosing a project will include the timescale, the size and variety of its team, its scale, and the likely loci of its major decision points. We will identify a group of 6-8 project team members for detailed case study, whom we will observe over a 6 month period approximately (depending on the exact timescale of the project). Team work and specialisation are essential to engineering practice, and this impacts on which people in a team are responsible for carrying out different types of mathematical analysis in a design project, and whether the mathematics involved is explicit or (to different degrees) packaged up into computer software. We will ensure that our case study subjects represent a range of specialisms. Regular team meetings will form the main focus for observation, because they are the principle points of interaction and discussion in the evolution of a design project, and will afford the opportunity to understand how mathematical meanings are negotiated by the different interests and parties involved in the project. This focus will also enable us to make the best use of time spent in situ, and to minimise any interference caused by our presence. We will supplement the meeting data with follow-up interviews with the case study subjects concerning any major decision points that occur during a meeting, that is where routine is disrupted by doubt and disagreement. In the TMO study, we identified “breakdown incidents” which proved especially valuable in exposing the normally tacit, embedded mathematical components of the subjects’ thinking, which are so important to examine, because these are precisely the mathematical elements that underlie the “intuitive” (and unquestioned) procedures that a professional uses in making routine judgements. In the current proposal, we prefer to call these “decision points” as they are likely to arise in discussion rather than in action — but the essential idea is similar. Decision points are also, of course, points at which

4

normally implicit mathematical (and professional) judgements have to be made explicit, and where the need to articulate decision criteria affords particular insight to observers. We will make regular update visits outside of team meetings, to all case study subjects during the 6 month observation period. We will collect copies of important documents and computer files where they figure in meetings and update visits. Phase 1 (Months 1-2): Documentary analysis We will begin by developing an overall audit of the general “visible mathematics” of engineering practice, that is, the mathematical elements, ideas and skills of engineering, which form the shared cultural assumptions of textbooks, academic courses, training materials and practitioners. We will take advantage of the extensive and explicit literature about engineering design and engineering mathematics (e.g. national and international accreditation standards for engineering education, specially-commissioned reports like Sutherland & Pozzi 1995, and books such as Addis 1990), and we will consult experts in this area where appropriate. Phase 2 (Months 2-3): Interviews at the engineering firm Phase 2 will involve the compilation and analysis of the particular “visible” mathematics of the firm’s engineering practice. We will interview a small number (4 or 5) of senior staff in the firm, in order to assist in the selection of a suitable project, and of the case study subjects. Selection of staff for interviews will be based on: (a) educational background, (b) age, and years of experience in the profession, and (c) work role within the practice—in particular, the amount of mathematical analysis involved in the job, and whether this is explicit or implicit in the use of different kinds of software. A project will be selected according to the criteria outlined above. We will then carry out preinterviews with the case study subjects to examine: (a) their mathematical background and attitudes towards mathematics: we will devise a suitable task to serve as the basis for this part of the interview; (b) their use of mathematics in their current work; (c) their expected role in the project. Our primary rationale for the interviews is to develop a clear view of the mathematics which practitioners admit to employing, in order that we can: a) examine if and how this mathematics is transformed as it is applied and negotiated in the context of the project and b) identify any other mathematical elements which emerge which are not predicted by the documentary or interview evidence of Phases 1 or 2. On the basis of all the interviews and documentary analysis, we will at this stage produce a provisional epistemological analysis of the mathematical applications the practitioners expect to occur during the project, and we will use this analysis to inform our observations during Phase 3 of the study. Phase 3 (Months 3-9): Ethnographic observation Tasks in this phase: •

Observation of meetings.

•

Post-meeting interviews with individuals to explore significant events and decision points during meetings.

•

Update visits to case study subjects, to monitor progress on the project, and to collect documents in progress. 5

Phase 4 (Months 7-11): Analysis and writing-up We will analyse the observational and interview data by constructing a series of episodes, that is contiguous pieces of data together with commentary which are delineated by an event or incident in the life of the project. Episodes will be based on fieldnotes, copies of resources, transcripts of interactions and interviews, and initially classified on the basis of the principal mathematical elements present within them. Each episode will comprise a description of the activity, the strategies used, the differences and similarities between different individual’s positions, and any other substantive issues that arise on close examination of the data. Using the episodes as a framework, case studies of the subjects will be developed whose main analytical focus will centre on how visible mathematics is applied and negotiated during the life of the project, and other mathematical skills and concepts which emerge as important but unpredicted by the subjects. The analyses of subjects’ reflections during the post-interviews will allow comparison between individual’s perspectives, and the extent to which particular mathematical practices are contingent on the roles, status, and professional expertise of individuals. Finally, a cross-sectional analysis of the case studies will attempt to draw theoretical conclusions from the empirical data. This will seek to elaborate how descriptions develop through webbing meanings from professional and mathematical knowledge, and to identify the situated abstractions articulated by the case study subjects during their project work. OUTCOMES OF THE PROJECT The main outcomes will be: 1. a set of analytical case studies of the mathematical work of professional civil engineers; 2. an elaboration of ongoing work on the relation between mathematical abstraction and professional expertise; 3. a theoretically and empirically grounded contribution mathematical training of engineering students.

to current debates on the

REFERENCES Addis, W. (1990). Structural Engineering: The nature of theory and design. London: Ellis Horwood. Czarnocha, B., Dubinsky, E., Prabhu, V. and Vidakovic, D. (1999). “One theoretical perspective in undergraduate mathematics education research”. Proceedings of the 23rd Psychology of Mathematics Education Conference, 1, 95-110. Hall, R. (1999). “Following mathematical practices in design-oriented work”. In Rethinking the Mathematics Curriculum, edited by C. Hoyles, C. Morgan and G. Woodhouse. London: Falmer Press. pp. 29-47. Hoyles, C., Noss, R. & Pozzi, S. (in press). “Proportional reasoning in nursing practice”. To appear in Journal for Research in Mathematics Education. IMA (1999) Engineering Mathematics Matters, Curriculum proposals to meet SARTOR 3 requirements for Chartered Engineers and Incorporated Engineers. The Institute of Mathematics and its Applications. Kent, P. (1999). Trajectories for Learning. Unpublished MA dissertation, Institute of Education, London.

6

Kent, P. & Noss, R. (2000). “The visibility of models: Using technology as a bridge between mathematics and engineering”. International Journal. of Mathematical Education in Science and Technology, 31, 1, 61-69. Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge University Press. Noss, R. and Hoyles, C. (1996). Windows on Mathematical Meanings: Learning Cultures and Computers. Dordrecht: Kluwer Academic Nunes, T., Schliemann, A. & Carraher, D. (1993). Street Mathematics and School Mathematics. Cambridge University Press. Pratt, D. & Noss, R. (submitted). “Illusions of contextualisation; Delusions of decontextualisation”. Submitted to Cognition and Instruction. Scribner, S. (1986). “Thinking in action: some characteristics of practical thought”. In R. J. Sternberg & R. K. Wagner (Eds.), Practical Intelligence: Nature and origins of competence in the everyday world. Cambridge, MA: Harvard University Press. Sutherland, R. and Pozzi, S. (1995). The Changing Mathematical Background of Undergraduate Engineers: A review of the issues. London: The Engineering Council. Tall, D. (Ed.) (1991). Advanced Mathematical Thinking. Dordrecht: Kluwer Academic. Tall, D. (1999). “Reflections on APOS theory in elementary and advanced numerical thinking”. Proceedings of the 23rd Psychology of Mathematics Education Conference, 1, 111-118.

7

1

BACKGROUND TO THE PROJECT In an earlier ESRC-funded study (1995-98), “Towards a Mathematical Orientation through Computational Modelling” (TMO), we studied the mathematical practices involved in the work of three groups of professionals—nurses, bankers and commercial pilots. In this new study, we want to build upon the results of TMO by looking at a group of professionals whose work involves a more complex repertoire of mathematical ideas, and we will take the specific example of civil engineers, observed at work in a large engineering consulting firm. The workplaces and the practitioners studied in TMO stand in contrast to those analysed in most previous studies of workplace mathematics, where researchers have sought to relate general kinds of “school” mathematics to “typical” everyday working practices—for example Scribner’s (1986) study of the dairy industry; Lave’s (1988) investigation of weight-watchers, and the work of Nunes et al (1993) on street vendors. In contrast, the TMO studies involved three professional groups which are similar in that they all have well-defined mathematical entry requirements developed through specific training programmes. These programmes usually try to provide students with somewhat general mathematical techniques that they will be able to apply in practice; yet one of the most notable observations in the study was the way in which these general techniques were not used in many elements of the practices that we observed, being instead replaced by a set of strategies dependent upon context. The TMO study sought to examine the epistemological and psychological issues involved in relating mathematical and professional knowledge. Moreover, the intention was to define mathematical knowledge in its broadest sense, and to include any activities which involved the mathematisation of workplace activity, not just those that consisted of fairly direct application of taught mathematical techniques. This allowed the study to gain a better understanding of how those essential mathematical elements focussed on in training were put to use in the realities of practice—and what other “mathematical” (in the sense of abstracted or generalised) understandings were evident. As a result, the TMO research has led to several detailed analyses which have shown that the various professional groups employed a variety of mathematical strategies which were finely tuned to their practice, yet simultaneously retained the notion of invariance that typifies abstract knowledge. For example, in the case of nurses, the invariant of “drug concentration” was found to underpin a set of proportional reasoning strategies to calculate drug dosages, tied to specific quantities and volumes of drugs, the way drugs are packaged and the organisation of clinical work (Hoyles, Noss & Pozzi, in press). From a theoretical point of view, these findings involve abstraction within a specific setting. Although abstraction has been a major concern of mathematics education research since the 1980s, most notably in the “advanced mathematical thinking” (AMT) tradition, research has focussed mostly on the learning of “pure” mathematics in university and post-compulsory schooling (see, for example, Tall 1991, 1999; Czarnocha et al 1999). In the past, we have criticised AMT’s view of the growth of mathematical understanding through a sequence of specific stages, and we have proposed (Noss & Hoyles, 1996) an alternative theoretical framework for abstraction in terms of •

webbing — a description of the ways in which learners come to construct new mathematical knowledge by forging and reforging internal (mental) connections through the interaction of internal and external resources during activity, and

•

situated abstraction — that learners construct mathematical ideas by drawing on the webbing of a particular setting which, in turn, shapes the way the ideas are expressed; that is, abstracting within, not away from, situations. 2

We intend this framework to be applicable in traditional mathematical learning situations, and recently we have looked in detail at how mathematical knowledge becomes abstracted in the contexts of 10-11 year olds’ construction of meanings for randomness (Pratt & Noss, submitted), and undergraduate-level applied mathematics (Kent, 1999). But we also intend that the framework should apply to situations where mathematics is being applied to other knowledge domains, and so, in the proposed study, we now want to extend our efforts further in the direction of situated abstractions of mathematics in professional work. RATIONALE OF THE PROJECT To build upon the results of the TMO study, and to continue to develop a theory of abstraction, we need to move the research into a domain of practice where we can see more sophisticated kinds of mathematics (geometry, algebra, calculus) that entail more complex structures of abstractions in their practical application. That is, we would like to investigate the problem of characterising mathematics in professional practices where mathematics plays a more overt, ubiquitous and complex role. In our chosen domain of civil engineering, we will be examining the process of engineering design, that is, devising a plan to carry out an engineering project before doing it. A design plan is typically very complex, and created over a long period of time—months or even years. Mathematical calculations, although crucial, form only one part of a set of results that support the final plan. Moreover, engineering design is done by teams of people (ranging in size from a handful to a hundred or more), made up of different kinds of specialists, where mathematical work may be the principle responsibility of only a small part of the team. We have a particular interest in the role of digital technologies in this collective design process: is mathematics becoming an ever more specialist realm, where non-specialists are increasingly just the consumers of “pre-cooked” mathematics hidden behind the interfaces of software packages? We believe that this question is not as clear cut as it seems, if we admit (as we did in TMO) a broader definition of mathematisation than the visible application of mathematical techniques. In the TMO study, we regarded the abstractions of mathematics into the professional practice as essentially single-layered: for example, a mathematical concept, such as ratio, is embedded into the practice of drug administration. We see engineering as a broad and important example of a more complex, mathematically-rich domain, where mathematical meanings and understandings are shaped, epistemologically and psychologically, by the activities of professional practice. Moreover, as Hall (1999) points out in the context of architectural design practice, mathematical meanings are socially negotiated in workplace settings, with multiple interests and different accountabilities driving the character of mathematical description within a given project. We intend “mathematically rich” in the sense not only of more mathematics, but also in the multi-layered structure of abstractions, in which a fragment of mathematical knowledge becomes embedded within another piece of mathematical, or scientific, or engineering knowledge—perhaps recursively. We propose to study how this multi-layered structure of engineering and mathematical knowledge becomes embedded into engineering practice. Relatedly, while there is no doubt that engineers use a great variety of mathematical concepts and techniques, this does not necessarily mean that it is mathematics in the (mathematician’s) sense of generalised, abstract knowledge. For example, in Kent & Noss (2000) we made a comparison of different mathematical software packages, some of which are marketed with a target audience of mathematicians and theoretical scientists, whilst others have a target audience of engineers. We think it is quite 3

intentional on the part of the software designers that the former type of package aims to present itself as a mirror to the mathematics knowledge domain, whilst the latter type treats mathematics as a “toolbox” of techniques for use in “applied” (engineering or scientific) problem solving. This comparison of perspectives suggests to us that mathematical epistemology is not a simple issue: it depends on what software designers, and users, think “mathematics” is. The answers to such epistemological questions have far from esoteric implications. The “application” metaphor underpins the mathematical training of undergraduate engineering students in the UK: in service mathematics courses, they are taught “the mathematics” that they will afterwards learn to apply to their engineering subject. There are clear signs that this curriculum system is coming under considerable strain: for example, two reports commissioned by the UK engineering institutions (IMA, 1999; Sutherland & Pozzi, 1995), have highlighted the problems caused by the significant changes to the teaching of mathematics in schools since the 1980s, with many students now lacking the kind of mathematical foundations which traditional service mathematics courses demand. We believe that our study of engineering practice will provide valuable information about the adequacy of the application metaphor as a basis for understanding how professional engineers use, and learn, mathematics, and that our findings will be of use to teachers of undergraduate engineers in developing revisions to service mathematics curricula. METHODOLOGY AND TIMETABLE We propose to take a broadly ethnographic approach which aims to analyse the material and social organisation of knowledge-in-use within a specific design project in a single large civil engineering firm. The early part of the study will involve gathering initial information, and negotiating with the firm the design phase of an engineering project for close and detailed observation. The criteria for choosing a project will include the timescale, the size and variety of its team, its scale, and the likely loci of its major decision points. We will identify a group of 6-8 project team members for detailed case study, whom we will observe over a 6 month period approximately (depending on the exact timescale of the project). Team work and specialisation are essential to engineering practice, and this impacts on which people in a team are responsible for carrying out different types of mathematical analysis in a design project, and whether the mathematics involved is explicit or (to different degrees) packaged up into computer software. We will ensure that our case study subjects represent a range of specialisms. Regular team meetings will form the main focus for observation, because they are the principle points of interaction and discussion in the evolution of a design project, and will afford the opportunity to understand how mathematical meanings are negotiated by the different interests and parties involved in the project. This focus will also enable us to make the best use of time spent in situ, and to minimise any interference caused by our presence. We will supplement the meeting data with follow-up interviews with the case study subjects concerning any major decision points that occur during a meeting, that is where routine is disrupted by doubt and disagreement. In the TMO study, we identified “breakdown incidents” which proved especially valuable in exposing the normally tacit, embedded mathematical components of the subjects’ thinking, which are so important to examine, because these are precisely the mathematical elements that underlie the “intuitive” (and unquestioned) procedures that a professional uses in making routine judgements. In the current proposal, we prefer to call these “decision points” as they are likely to arise in discussion rather than in action — but the essential idea is similar. Decision points are also, of course, points at which

4

normally implicit mathematical (and professional) judgements have to be made explicit, and where the need to articulate decision criteria affords particular insight to observers. We will make regular update visits outside of team meetings, to all case study subjects during the 6 month observation period. We will collect copies of important documents and computer files where they figure in meetings and update visits. Phase 1 (Months 1-2): Documentary analysis We will begin by developing an overall audit of the general “visible mathematics” of engineering practice, that is, the mathematical elements, ideas and skills of engineering, which form the shared cultural assumptions of textbooks, academic courses, training materials and practitioners. We will take advantage of the extensive and explicit literature about engineering design and engineering mathematics (e.g. national and international accreditation standards for engineering education, specially-commissioned reports like Sutherland & Pozzi 1995, and books such as Addis 1990), and we will consult experts in this area where appropriate. Phase 2 (Months 2-3): Interviews at the engineering firm Phase 2 will involve the compilation and analysis of the particular “visible” mathematics of the firm’s engineering practice. We will interview a small number (4 or 5) of senior staff in the firm, in order to assist in the selection of a suitable project, and of the case study subjects. Selection of staff for interviews will be based on: (a) educational background, (b) age, and years of experience in the profession, and (c) work role within the practice—in particular, the amount of mathematical analysis involved in the job, and whether this is explicit or implicit in the use of different kinds of software. A project will be selected according to the criteria outlined above. We will then carry out preinterviews with the case study subjects to examine: (a) their mathematical background and attitudes towards mathematics: we will devise a suitable task to serve as the basis for this part of the interview; (b) their use of mathematics in their current work; (c) their expected role in the project. Our primary rationale for the interviews is to develop a clear view of the mathematics which practitioners admit to employing, in order that we can: a) examine if and how this mathematics is transformed as it is applied and negotiated in the context of the project and b) identify any other mathematical elements which emerge which are not predicted by the documentary or interview evidence of Phases 1 or 2. On the basis of all the interviews and documentary analysis, we will at this stage produce a provisional epistemological analysis of the mathematical applications the practitioners expect to occur during the project, and we will use this analysis to inform our observations during Phase 3 of the study. Phase 3 (Months 3-9): Ethnographic observation Tasks in this phase: •

Observation of meetings.

•

Post-meeting interviews with individuals to explore significant events and decision points during meetings.

•

Update visits to case study subjects, to monitor progress on the project, and to collect documents in progress. 5

Phase 4 (Months 7-11): Analysis and writing-up We will analyse the observational and interview data by constructing a series of episodes, that is contiguous pieces of data together with commentary which are delineated by an event or incident in the life of the project. Episodes will be based on fieldnotes, copies of resources, transcripts of interactions and interviews, and initially classified on the basis of the principal mathematical elements present within them. Each episode will comprise a description of the activity, the strategies used, the differences and similarities between different individual’s positions, and any other substantive issues that arise on close examination of the data. Using the episodes as a framework, case studies of the subjects will be developed whose main analytical focus will centre on how visible mathematics is applied and negotiated during the life of the project, and other mathematical skills and concepts which emerge as important but unpredicted by the subjects. The analyses of subjects’ reflections during the post-interviews will allow comparison between individual’s perspectives, and the extent to which particular mathematical practices are contingent on the roles, status, and professional expertise of individuals. Finally, a cross-sectional analysis of the case studies will attempt to draw theoretical conclusions from the empirical data. This will seek to elaborate how descriptions develop through webbing meanings from professional and mathematical knowledge, and to identify the situated abstractions articulated by the case study subjects during their project work. OUTCOMES OF THE PROJECT The main outcomes will be: 1. a set of analytical case studies of the mathematical work of professional civil engineers; 2. an elaboration of ongoing work on the relation between mathematical abstraction and professional expertise; 3. a theoretically and empirically grounded contribution mathematical training of engineering students.

to current debates on the

REFERENCES Addis, W. (1990). Structural Engineering: The nature of theory and design. London: Ellis Horwood. Czarnocha, B., Dubinsky, E., Prabhu, V. and Vidakovic, D. (1999). “One theoretical perspective in undergraduate mathematics education research”. Proceedings of the 23rd Psychology of Mathematics Education Conference, 1, 95-110. Hall, R. (1999). “Following mathematical practices in design-oriented work”. In Rethinking the Mathematics Curriculum, edited by C. Hoyles, C. Morgan and G. Woodhouse. London: Falmer Press. pp. 29-47. Hoyles, C., Noss, R. & Pozzi, S. (in press). “Proportional reasoning in nursing practice”. To appear in Journal for Research in Mathematics Education. IMA (1999) Engineering Mathematics Matters, Curriculum proposals to meet SARTOR 3 requirements for Chartered Engineers and Incorporated Engineers. The Institute of Mathematics and its Applications. Kent, P. (1999). Trajectories for Learning. Unpublished MA dissertation, Institute of Education, London.

6

Kent, P. & Noss, R. (2000). “The visibility of models: Using technology as a bridge between mathematics and engineering”. International Journal. of Mathematical Education in Science and Technology, 31, 1, 61-69. Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge University Press. Noss, R. and Hoyles, C. (1996). Windows on Mathematical Meanings: Learning Cultures and Computers. Dordrecht: Kluwer Academic Nunes, T., Schliemann, A. & Carraher, D. (1993). Street Mathematics and School Mathematics. Cambridge University Press. Pratt, D. & Noss, R. (submitted). “Illusions of contextualisation; Delusions of decontextualisation”. Submitted to Cognition and Instruction. Scribner, S. (1986). “Thinking in action: some characteristics of practical thought”. In R. J. Sternberg & R. K. Wagner (Eds.), Practical Intelligence: Nature and origins of competence in the everyday world. Cambridge, MA: Harvard University Press. Sutherland, R. and Pozzi, S. (1995). The Changing Mathematical Background of Undergraduate Engineers: A review of the issues. London: The Engineering Council. Tall, D. (Ed.) (1991). Advanced Mathematical Thinking. Dordrecht: Kluwer Academic. Tall, D. (1999). “Reflections on APOS theory in elementary and advanced numerical thinking”. Proceedings of the 23rd Psychology of Mathematics Education Conference, 1, 111-118.

7