Designing Learning Opportunities for
Techno-mathematical Literacies in
Financial Workplaces: A status report
Arthur Bakker, Celia Hoyles, Phillip Kent and Richard Noss
Institute of Education, University of London
technomaths@ioe.ac.uk
www.ioe.ac.uk/tlrp/technomaths
Paper presented at the 6th Annual Conference of the Teaching and Learning
Research Programme, Warwick, 28-30 November 2005
DRAFT – NOT TO BE QUOTED
ABSTRACT. The “Techno-mathematical Literacies in the Workplace” project is
investigating the needs of employees in a range of industrial and commercial
workplaces to have functional mathematical knowledge that is grounded in their
workplace situations and in the technological artefacts that surround them. We
describe this kind of knowledge in terms of “Techno-mathematical Literacies” (TmL),
and we report in this paper on our developing ideas for training in TmL, which we call
“learning opportunities”: flexible resources for mathematical learning that can be
incorporated within workplace technical training materials, and are situated within the
contexts and artefacts of the workplace. Where mathematical skills are required in
workplaces, the training of de-contextualised mathematical ideas has long been
recognised as a problematic approach to effective skills development. We will
explain how our situated approach may contribute to more effective training practices.
We will describe the development process, and present an example of a learning
opportunity in the mortgage sales area of the financial services industry, in which we
are addressing the needs of sales agents to develop a better understanding of the
mathematical models which underpin the complex mortgage product that they deal
with.
On the basis of this example, we will discuss several key ideas for our development
work: (1) “situated modelling”, that is, the need for employees to develop models of
processes that allow them to make data-informed decisions and to communicate
with colleagues, managers and customers about the systems and processes that
they work with; (2) the role of “boundary objects” that mediate workplace practices
and hence can be used to develop workplace learning.
KEYWORDS: Knowledge at work, Learning contexts and settings, Learning
trajectories, Mathematics, Computers/IT, Boundary objects
-1-
Introduction
The “Techno-mathematical Literacies (TmL) in the Workplace Project” is
investigating the combinations of mathematical, statistical and technological skills
that people need in workplaces. We are investigating three contrasting industry
sectors (Pharmaceuticals Manufacturing, Packaging, Retail Financial Services 1) and
we focus on employees at “intermediate” skill level, typically non-graduates with Alevel (or equivalent) qualifications who may be working in service industry (such as
banking) as sales agents or customer enquiry agents, or in manufacturing industry
as skilled operators or supervisory managers.
In a previous project (Hoyles et al, 2002) we promoted the idea of “mathematical
literacy” as a growing necessity for successful performance in the workplace. In the
current project, we are using the term “Techno-mathematical Literacies” (TmL) as a
way of thinking about mathematics as it exists in current IT-based workplace
practices. The idea of literacy is crucial: individuals need to be able to understand
and use mathematics as a language that will increasingly pervade the workplace
through IT-based control and administration systems as much as conventional
literacy (reading and writing) has pervaded working life for the last century.
As workplaces become increasingly organised around IT systems, the tendency is
for mathematical processes to become less visible when the mathematics becomes
performed by IT. In such cases, the nature of the mathematical skills required
changes: there is a shift in requirement from fluency in doing explicit “pen and paper”
mathematical procedures to a fluency with using and interpreting output from IT
systems and software, and the mathematical models deployed within them, to carry
out mathematical procedures in order to inform workplace judgements and decisionmaking. Significantly, the mathematical skills shift to a broader form: whereas penand-paper techniques may have been for most employees a matter of narrowly
following through a calculation without having to worry about the context or meaning
of that calculation, the new emphasis is around using calculations appropriately in
contexts.
It is this kind of broader thinking that we intend with the notion of “technomathematical literacies” (TmL), with the “techno-mathematics” part emphasising that
the visible mathematics of workplaces, the mathematics recognised in formal training
procedures, is only part of a more broadly-defined and technologically-shaped
mathematics, which remains largely invisible and embedded within the routines of
working practice.
The project’s research consists of two phases. The first (described in our paper for
the previous TLRP Conference: Bakker et al, 2004) was concerned with identifying
the mathematical practices which are present in the three industry sectors. This
phase involved case studies in three or four companies per sector. Based on workshadow observations and interviews, we aimed to understand the work process and
to describe what was techno-mathematical about the practices that we observed.
1
We have recently begun developing learning opportunities in a fourth sector, automotive
manufacturing, which shares many characteristics with pharmaceuticals manufacturing.
-2-
One of the results is a set of real contexts and situations in which we think
employees’ TmL can be improved.
The current phase of the research is thus concerned with the question of how we
can support employees in developing the TmL that are useful in their work. We carry
out design experiments (Cobb et al, 2003) in collaboration with companies and
industry sector experts, which are characterised by design cycles of preparing,
designing, testing and revision of materials that we call learning opportunities. These
are flexible resources for mathematical learning that will eventually be incorporated
within, or be presented alongside, workplace technical training materials. During the
process of development, we continue to learn from the learners, trainers and
managers within companies. We propose “learning” over “training” to emphasise that
rather than thinking of training as transmitting our mathematical knowledge to
employees and managers in companies we think of learning opportunities as
potentially “boundary crossing” activities (see below) involving the participants and
ourselves, essentially connecting mathematical knowledge and work process
knowledge.
This implies we take employees’ perspectives seriously and thus we use learning
opportunities as “windows” onto their thinking. The learning opportunities are
carefully designed to weave mathematical ideas into real situations using appropriate
technology in a constructionist way and are based around problems that we have
observed in workplaces, so as to facilitate rich discussion. We devise activities
where employees are given space to voice the meanings they bring to mathematical
artefacts such as graphs and mathematical concepts, even if these are seen formally
as incorrect. Unpacking why they attribute possibly “incorrect” meanings to artefacts
is an important element of the learning opportunity, both to them and to us. We see
negotiation around boundary objects as an important step in employees’ articulation
of TmL, so that they may develop “situated abstractions” where mathematical tools
both frame their understanding and are the means by which they are communicated
(cf. Noss & Hoyles, 1996).
In this paper we will describe an example of a “skills gap” that we have observed in a
financial workplace concerning TmL for the financial mathematics involved in
mortgage products, and we will present some emerging ideas (as yet, very early
prototypes) for learning opportunities intended to address this situation.
A methodological note
Like most researchers in education who adopt ethnographic methods, we are not
able to attempt the kind of engagement which is typical of ethnography amongst
professional anthropologists (immersion of the researcher in the community under
investigation). We can spend only short periods of time in workplaces, and so to
guard against the problem of biased observation we place significant effort on
triangulation. In collecting data, we continuously seek to triangulate different views of
the same workplace activity, through the perspectives of employees at all levels. In
analysing data, we triangulate interpretations of the raw data (audio transcripts,
photographs of workplaces, artefacts in the form of paper documentation) amongst
the project team. We further triangulate our findings by appealing to experts from the
particular industrial sector, by means of direct consultation and through validation
-3-
meetings in which sector experts are invited to learn about project findings, comment
on their validity and generality, and suggests ways forward for the research. The
triangulation of research findings effectively continues through the cycles of the
design experiments as we gain feedback from the prototyping of materials which
serves to refine and extend the original findings.
Analytical framework: Workplaces as activity systems
We will briefly outline our theoretical framework for analysing mathematical practices
in workplaces. We seek to understand how different companies deploy IT-based
systems, the forms of (mathematical) knowledge required by employees to operate
effectively and how these relate to the managerial strategies adopted by a company.
The basic premise of activity theory is helpful in understanding the role of TmL in
workplaces: that people work to realise an object of activity (i.e. the purpose of work)
through actions that are mediated by artefacts, for example computers and the
information that they provide. We interpret each workplace as a complex
arrangement of interacting activity systems each characterised by its own object,
mediated by artefacts and located in a context characterised by a specific division of
labour, sets of rules and inter-related workplace communities (see, for example,
Kuutti, 1996; Engeström, 2001).
Our thinking about how to conceptualise the relations between the objects of activity
and the actions carried out by individuals, both within and between activity systems,
is influenced by the debate in the activity theory literature about “boundary-crossing”
and “boundary objects” (Tuomi-Gröhn and Engeström, 2003). Boundary crossing
builds on Star and Griesemer’s (1989) notion of a boundary object, an object which
serves to coordinate different perspectives of several communities of practice.
Boundary objects are flexible enough that different social worlds can use them
effectively and robust enough to maintain a common identity among those worlds.
Boundary crossing happens if boundary objects are used across the boundaries of
different activity systems, or between different communities within an activity system,
in ways that facilitate communication between and within systems. Then, tacit
knowledge and assumptions can be made more explicit and individuals from
different communities can learn something new.
This idea is the basis for our approach to the learning of TmL in workplaces, seeking
to develop boundary objects and boundary crossing situations based on authentic
workplace artefacts and situations, in order to create new knowledge across the
boundaries.
Techno-mathematical Literacies in financial workplaces
A significant feature of TmL as we have seen it in financial services companies is
that the mathematics involved seems not much different from what appears in the
secondary school mathematics curriculum (for example, calculating compound
interest). Yet the effect of workplace context is to introduce a significant degree of
complexity to even the simplest mathematics, since any mathematical procedure is
not an isolated mathematical exercise but is part of a set of decisions and
judgements that have to be made about what is generally a complex process.
Moreover, this complexity is, for the most part, hidden in the IT models, only
-4-
unexpectedly surfacing when customers ask difficult questions or different
communities need to communicate across knowledge boundaries.
Through our dialogues with financial services companies, we have become
convinced that our suppositions about TmL and their importance for workplaces are
broadly correct. We have further noted a general pattern of “maths avoidance” that
runs through training and working practices. Companies are reorganising to deal with
changed circumstances, and the new skills requirements are being recognised, but it
seems these are often not responded to in a positive way. For example, in the
company we are about to describe we were told that mathematical issues are
deliberately avoided in initial training for sales employees because it would frighten
and alienate many of the trainees. The situation is, however, that employees do
become generally functional through this training, and indeed the company, because
of its niche product market, enjoys good profitability. Yet a narrowly-trained
workforce is less effective against the uncertainties of the future and will lack
flexibility to change. Increasingly, the competitive environment and an increasingly
knowledgeable customer base (thanks to the internet) are restricting the
effectiveness of functional product knowledge. It is for breadth of understanding
about “where the numbers come from” and flexibility for changing practice that we
advocate the importance of TmL, including individual needs for personal and career
development.
An example of TmL: Selling current account mortgages
Important examples of TmL that we have found ubiquitous in financial services are:

appreciating models of financial products and their inter-related outputs;

reading graphs and being able to communicate the meanings of graphs in
context.
An example which combines these 2 comes up in the selling of current account
mortgages, which we have investigated in a specialist mortgage provider. The key to
selling here is to establish how the current account mortgage, which is extremely
flexible but cannot offer a low interest rate, is different to the majority of mortgages
which require fixed regular repayments in return for a discounted interest rate.
Prospective customers typically seek out information by making a phone call to a
sales agent (customers can do essentially the same for themselves by using the
company website). The agent inputs personal and financial details, as well as the
property for which the customer is seeking a mortgage, into the computer system. If
the basic lending criteria are satisfied, the agent then goes into a typically rather long
computer-assisted dialogue (20 minutes or more) which leads to the generation of a
printed illustration document based on how the customer might use the current
account mortgage: how much would he pay in each month as salary, how much
savings can be left in the account, any regular bonuses from his employment, any
2
The financial services work that we have seen features surprisingly little use of graphical
representations, with numbers, often in tabular form, being very much dominant. This contrasts with
science and engineering practice (including school mathematics and science) which is extremely rich
in the use of graphical representations. Although IT is ubiquitous in financial services, it seems that it
has not yet changed the rather a-graphical culture of working practices.
-5-
outstanding credit card debts or loans that can be combined with the mortgage, and
so on.
Each of these inputs leads to possible savings on the cost of the mortgage since the
mortgage debt is “offset” against income and savings – that is, interest is paid only
on the difference between the outstanding mortgage and the positive balance held in
the account. A standard “persuasive graphic” that is used shows the outstanding
balance of the mortgage over time, and the effects on the graph of offsetting – see
Figure 1 below which shows the standard repayment of £100,000 borrowed over 20
years (grey line), compared with offsetting £20,000 of savings for the life of the
mortgage (black line). By saving on interest (more than £28,000 over the whole term),
capital can be repaid more quickly, saving in the case shown several years on the
“standard” mortgage term, assuming that the same monthly payment is made
regularly. (This is an overly simplistic case, but illustrates the effect of offsetting; in
fact, most users of a current account mortgage make quite variable repayments,
overpaying and underpaying with respect to the “standard” amount as their financial
circumstances change.)
100000
90000
Outstanding balance £
80000
70000
60000
50000
40000
30000
20000
10000
0
0
5
10
15
20
Years
Figure 1: Outstanding balance graphs and “savings” information for a current
account mortgage
Graphs and data like this (generated by “black box” software) are a key tool used in
selling current account mortgages, where a central point to be made to the customer
is that discounted interest rates are not the only things that matter about a mortgage,
contrary to the way that most mortgages are sold. Common difficulties with this
information are:

the customer is quoted a saving in money and years of repayment which often
sounds “too good to be true”; in the absence of any visible calculation there
are few grounds on which the customer or the sales agent has a basis with
which to quantify the reported savings – the sales agent cannot make a
“common sense” judgement as to whether the figures are realistic or how to
bring them to life and make them meaningful to the person on the phone;

sales agents are trained to follow a standard script that is tuned to the
behaviour of the majority of customers. The minority who ask unusual
questions, or who do not wish to follow through the standard long script for
illustrations, are likely to find the sales agent “floundering” (the expression of
-6-
one of the training managers) and unable to answer the customer’s question
beyond vague generalities.
Given the importance of the repayment graphs, we investigated how sales agents
understand them. We found that none had more than a superficial idea of how the
graphs are calculated by the software. In itself this is not surprising. It raises the
question of whether it matters that the user of the tool does not know what the
creator of the tool has done? One reason why it might, is that there are key
parameters for the mortgage that are not directly visible in the graph, and hence not
connected to the displayed numerical information – among these are the interest rate,
the monthly payment, and the possible tax benefits for a higher-rate taxpayer.
Moreover, there would be a benefit in knowing something about how the parameters
inter-relate to determine the repayment characteristics, since a significant problem
reported to us by managers is that sales agents could have a better appreciation of
how the different aspects of the current account mortgage fit together.
Communicating attractive features of the mortgage involves a network of reasoning
steps by which agents are able to link a customer concern or characteristic to a
feature or benefit of the product.
In suggesting an improved reasoning ability with the graphs and other numerical
information, we are definitely not advocating mini “algebra refresher courses” for
sales agents as a basis for improved understanding. Rather, there is a need for
employees to explore the relationships that are embedded in the model of the
mortgage that is encoded in the software – to reason about these relationships, and
their effects on “outputs” (mortgage repayments), and having a language to
communicate these to customers.
Learning opportunities
We are currently developing learning opportunities that will address the situation of
current account mortgages described above. Key aspects to our work on learning
opportunities are

use of realistic mathematical artefacts, as a basis for the creation of boundary
objects;

learning needs to happen in the context of the work with the aim to develop a
situated appreciation of the analytical models in use, as expressed through
the IT systems;

use of realistic “scenarios” (actual customer interactions and how they were
dealt with): especially “surprise” situations which motivate a need for
explanation in quantitative, techno-mathematical terms;

“open” software, such as spreadsheets, which allow learners to construct
formal mathematical ideas for themselves (in line with the constructionist
approach to learning – cf. Noss & Hoyles, 1996), so that learners can build
models of financial products and situations, opening up the closed
calculations of the IT system. These offer a dynamic picture of a situation,
where “input” quantities can be changed and the effects on outputs can be
observed;

collaborative activities where learners
communicate their results to others;
-7-
have
to
work
together
and

interventions which integrate with existing training/employee development
modes, and involve managers at different levels so that there is “buy-in”.
Working within the design experiment paradigm, we work on design cycles of
preparing, designing, testing and revising materials in collaboration with companies
and (where accessible) industry sector experts. The aim is that learning opportunities
will eventually be incorporated within workplace technical training materials, and so it
is essential that the company has ownership of them. An important aspect of the
boundary crossing activity involving the participants and ourselves is therefore to
develop this sense of ownership.
We are developing learning opportunities by starting from the actual products and
systems in use in a company, and trying to open up some of the calculations and
decisions which are carried out as black boxes behind computer screens. A number
of such calculations occur for current account mortgage illustration, particularly
concerning the results of offsetting savings against debts.
Our general aim here is to take mathematical artefacts that learners know as
everyday objects in the workplace – such at the repayment graph illustrated above –
and make them the focus of a discussion between learners, and between learners
and ourselves – that is, to become boundary objects for shared communication and
understanding. As we have suggested above, training and practice in this company
tends to “steer around” the mathematics of such artefacts. Thus the idea of
discussing how these artefacts work mathematically may be a very novel experience
with such familiar objects. The power of discussion is twofold: it prompts learners to
ask their own questions about the objects, and therefore to seek mathematical
answers using the software; secondly, it helps us to find out how learners think.
Repeatedly, we find ourselves making assumptions about what people will think of
the objects and tasks that we present – assumptions which reflect our own biased
viewpoint as mathematicians. The point is that meanings are strongly rooted in
contexts, and as outsiders to the workplace context we continually have to reevaluate our interpretations as we learn more about the context.
We are using open software (Excel spreadsheets) in the learning opportunities which
seek to open up “black box” calculations for investigation, and which allow for
exploration and manipulation of variables in calculations. Figure 2 shows a sample
screenshot, illustrating a spreadsheet model for mortgage repayments, resulting in
repayment graphs equivalent to those shown in Figure 1; all the input variables and
calculations are made explicit here and the inputs may be varied to see the effect on
the repayment graphs. Note here that the whole structure is potentially modifiable by
simple changes that lead to models of more complex situations; this is an important
aspect of what we mean by “open-ness” of the software. Where possible, we invite
learners to build their own spreadsheets, and then modify them bit-by-bit to model
more realistic situations.
Spreadsheets like the one shown in Figure 2 are supplemented by specially-written
macros and applications to provide special financial functions (that would otherwise
require complex formulae to appear in the spreadsheet), and to provide interfaces for
doing calculations where we want learners to focus on specific relationships (by
varying one or more quantities – e.g. by slider controls – and looking for the effect on
outputs) and not worry about calculation details.
-8-
Figure 2: Screenshot from a learning opportunity using Excel to model mortgage
repayments; the input variables to the model (mortgage term, interest rate, offset
savings) are made explicit and modifiable, and the results of changes can be seen
directly in the repayment graphs (equivalent to those shown in Figure 1).
Calculations on the right (no offsetting) generate the grey line, and calculations on
the left (with offsetting against savings) generate the black line.
We do not attempt to bring into the foreground all the “essential” mathematical
details. Rather, we let the software tool mediate the mathematics, and the
spreadsheet in many situations can allow for an approach that keeps algebraic
formalism “encoded” within the spreadsheet structure in such a way that the learner
can see rather clearly what the formalism is doing. This is in line with our overall
emphasis on guiding learners to use mathematical ideas appropriately, rather than
guiding them to do the explicit mathematical calculations involved.
To develop this point further, we have proposed a set of techno-mathematical
literacies which make up a collective literacy that we term situated modelling (Bakker
-9-
et al, in preparation). We seek to contrast “mathematical modelling” as it is
conventionally understood with a description of modelling that fits with how Technomathematical Literacies may operate in the workplace. Situated modelling involves
the activities of “making the invisible visible” in the form of “situated models” and
using those models as the basis for decision-making. We characterise these models
as situated because their understanding depends on a combination of contextual
and mathematical issues. This kind of modelling is rather different from the standard
idea of mathematical modelling in which a real-world situation is “translated” into a
mathematical model so that it can be used to solve a mathematised problem or
make a prediction that is translated back to the real world, while neglecting any
“noise” from the context 3 . The situated approach takes the noise (from a
mathematical perspective) as providing much of the meaning to the model.
Conclusions
We have presented in this paper the current status of our research on analysing
mathematical practices in financial workplaces, and developing learning
opportunities for Techno-mathematical Literacies that seek to respond to employees’
needs to reason more effectively with mathematics, in response to business needs.
We illustrated this with an example of a mortgage provider which despite current
good profitability is aware that its sales employees are rather narrowly-trained for
future needs. However the route towards a broader training requires an engagement
with the mathematical ideas underlying financial products, which this company
presently avoids so as not to alienate most of its sales employees.
In our response to this kind of situation, we have emphasised the changing nature of
mathematical practices in workplaces, arising from the introduction of IT. This is a
point which we feel is not sufficiently recognised (at least in the UK) in education and
workplace-related research and in policy discussion around mathematics and
numeracy.
It is interesting to reflect on the learning opportunities that we have created at the
current early stage of our design experiment. They are quite “mathematical” in
flavour, despite our continual attempt to engage with realistic contexts. Actually, we
think this transition is inevitable, in the process of boundary crossing between
ourselves and the company involved. We have adopted this methodology, on the
basis of previous research experience, as a way of dealing with the perennial
“problem of transfer” of de-contextualised mathematical knowledge into everyday
workplace practice (cf. Tuomi-Grohn & Engestrom, 2003). Our approach also gives
recognition to the fact that employees develop through experience very rich and
generally tacit understandings and ways to cope with the complexities of their
workplace situations and these understandings are to some extent mathematised.
Mathematical practices at work are inseparably tied up with everyday activity at work,
the tools used and how they are used to solve business problems and communicate
relevant mathematical ideas. The challenge for the development of learning
materials is to develop connections and an enhanced discourse across the boundary
between workplace and mathematical communities.
“Mathematicians are like Frenchmen: whatever you say to them they translate into their own
language and forthwith it is something entirely different.” (Goethe, 1829)
3
- 10 -
Acknowledgement
Funding of this research project October 2003 – March 2007 by the ESRC Teaching
and Learning Research Programme [www.tlrp.org] is gratefully acknowledged
(Award Number L139-25-0119).
References
Bakker, A., Hoyles, C., Kent, P. and Noss, R. (2004, November). Technomathematical Literacies in the workplace: Improving workplace processes by
making the invisible visible. Paper for the TLRP Annual Conference 2004 [Online:
www.tlrp.org/dspace/handle/123456789/199 ]
Bakker, A., Hoyles, C., Kent, P. and Noss, R. (in preparation). “Improving work
processes by making the invisible visible and coming to data-informed decisions”.
Cobb, P., Confrey, J., diSessa, A. A., Lehrer, R., & Schauble, L. (2003). “Design
experiments in educational research”. Educational Researcher 32, 9-13.
Engeström, Y. (2001). “Expansive learning at work: toward an activity theoretical
reconceptualization”. Journal of Education and Work, 14, 1, 133 – 156.
Hoyles, C., Wolf, A., Molyneux-Hodgson, S. and Kent, P. (2002), Mathematical Skills
in the Workplace. London: Science, Technology and Mathematics Council.
[Online: www.ioe.ac.uk/tlrp/technomaths/skills2002 ]
Kuutti, K. (1996). “Activity theory as a potential framework for human-computer
interaction research”. In Nardi, B. (ed.), Context and Consciousness: Activity
Theory and Human-computer Interaction, pp. 17-44. MIT Press.
Noss, R. & Hoyles, C. (1996) Windows on Mathematical Meanings: Learning
cultures and computers. Dordrecht: Kluwer Academic.
Star, S. L. (1989). “The structure of ill-structured solutions: Boundary objects and
heterogeneous distributed problem-solving”. In L. Gasser & M. N. Huhns (Eds.),
Distributed Artificial Intelligence, volume 2. London: Pitman / San Mateo, CA:
Morgan Kaufmann.
Star, S. L. and Griesemer, J. (1989). “Institutional Ecology, ‘Translations,’ and
Boundary Objects: Amateurs and professionals in Berkeley’s Museum of
Vertebrate Zoology, 1907-1939,” Social Studies of Science, 19: 387-420.
Tuomi-Gröhn, T. and Engeström, Y. (Eds.) (2003). Between School and Work: New
Perspectives on transfer and boundary crossing. Oxford: Pergamon.
- 11 -