# Boundary crossing between school and work for developing techno-mathematical competencies in vocational education

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## General

In a high-tech knowledge-based economy many employees need Techno-mathematical Competencies (TmC): these are combinations of technological and mathematical competencies that are mediated by technology and situated in specific work contexts. At the boundaries between vocational colleges (mbo) and learning-work sites, the proposed project focuses on the question: How can new learning arrangements support the development of the TmC required for technical work? Among the results will be a task-specific instruction theory and computer tools for developing such TmC. The theoretical focus is on boundary crossing between school and work as a tool to promote progressive recontextualisation of techno-mathematical knowledge.

## Problem definition

### Boundary crossing between school and work

A major challenge for students in vocational education and training (VET) is to make connections between school and work settings. For example, as Van der Sanden and Teurlings (2003) observe, even though the vocational school system aims at connecting students’ codified knowledge acquired at school to episodic knowledge developed from workplace practice periods, this often does not lead to the required integration of different knowledge forms. The boundary between school and work is therefore an interesting location to study what is metaphorically called transfer. Traditionally, general subjects such as mathematics were (and often still are) taught in a general way assuming that the knowledge acquired during education can later be applied in specific contexts, but there are two problems with this view on transfer.

Firstly, this view has been criticised by many proponents of situated cognition, constructivism and socio-cultural theories of learning for its lack of attention for context, participation in communities, history and available tools (e.g., Beach, 1999; Lave, 1988; Van Oers, 1998). Secondly, the implied abstract approach makes general subjects such as mathematics and science difficult for many students and this is an important cause for school failure (Onstenk, 2002). These are among the reasons that Dutch VET has moved to a competency-based approach in which general subjects are increasingly integrated into practical tasks, for example by using a problem-centred approach. Although positive examples are reported (Van der Sanden, 2004), communication between school and work is characterised as mostly uni-directional from school to work sites, and concerns are expressed that too much emphasis on practical learning leads to decreasing quality of education and disappearance of general subjects (e.g., SCP, 2006).

From an educational-theoretical point of view there is a pressing need for a theoretical basis of how to characterise the required competencies and how to integrate relevant knowledge forms. Because transfer always involves a learning process in the new situation, Van der Sanden and Teurlings (2003) propose to focus on “continuous progressive recontextualisation” (cf., Guile & Young, 2003; Van Oers, 1998). When entering new situations, students always have to recontextualise their knowledge and when supported in reflecting on their experiences and progressively organising their knowledge, they improve relevant competencies. An important question is then how to promote this process of recontextualisation.

Tuomi-Gröhn and Engeström (2003) argue that one way is by “boundary crossing” between different activity systems, for example at school and at work. An activity system can be seen as the smallest unit of analysis that takes into account the purpose of work, available tools, division of labour and the rules of discourse in workplace communities. In transitions from one system to another, students as well as their teachers and practical supervisors take part in different communities, which have different goals, tools and sometimes even opposite interests (SCP, 2006). More concretely, Onstenk (2003) and Van der Sanden (2004) offer suggestions on how the required boundary crossing can be achieved. For example, teachers should visit workplaces more often and school tasks should more often be related to the core tasks (in the sense of Onstenk; 2003) of the profession. Designing a powerful learning environment thus involves the boundary crossing by many participants such as college teachers, practical work supervisors and technical expertise centres.

### Techno-mathematical Competencies

Though recent developments in educational research such as the activity-theoretical approach have brought an interesting perspective on the issue of transfer in VET, Guile and Young (2003, p.79) observe that “the role of scientific concepts seems to have got lost in recent developments in activity theory with their stress on activities, context and horizontal development”. This proposal will therefore address the role of technical and mathematical knowledge in workplaces, which involves vertical development and some level of generalisation, while taking into account recent insights from educational-theoretical approaches.

As such the proposed project builds upon ESRC-funded research that has been carried out in the UK, in particular the Techno-mathematical Literacies in the Workplace project (www.ioe.ac.uk/tlrp/ technomaths) in which the candidate for this proposal has been a research officer for two years. Techno-mathematical Literacies (TmL) can be characterised as combination of technical and mathematical competencies, mediated by technology available in work situations (Bakker et al., 2006). In the context of Dutch competency-based VET we use the term TmC (Techno-mathematical Competencies) instead of TmL (Literacies). In line with Van der Sanden (2004) and others, competencies are characterised as conglomerates of knowledge, skills and attitudes required to carry out particular professions.

The need for TmC is apparent in labour-market survey studies (e.g., Felstead et al., 2002) and in case studies of workplaces (Bakker et al., 2006; Bessot & Ridgway, 2000). This is mainly due to the increasing use of IT at work: instead of using a spanner, an operator might have to use data and graphs in a control panel to fix a production problem. A detailed study of skills levels and needs of operational and supervisory staff in life sciences shows that among the most problematic and yet important skills are those “concerned with understanding and use of Statistical Process Control (SPC), monitoring use of SPC techniques during routine production, monitoring data.” (MerseyBio, 2006, p. 46). To substantiate this finding we give one example from the TmL project (Hoyles et al., in press).

### An example of TmC

Many industrial companies use statistical control charts, which display key measures of manufactured items produced in relation to a target value, statistical control limits and specification limits. The control limits are defined such that 99.7% of the common cause variation stays within those limits (mean +/-3 standard deviations) whereas special cause variation can be seen as trends, patterns or as data points outside the limits. Operators are expected to monitor the process and spot anything deviating from random, common cause variation. Particularly if software is used, these control charts are often experienced as black boxes that do things users are not aware of. For example, software packages may ignore outliers in their calculations of control limits. To make data-informed decisions, it is therefore crucial to know some of the statistics embedded in these tools as well as features of the technology itself. Despite the importance of SPC in most business improvement programmes as used by companies such as Philips, Douwe Egberts and ABN Amro, it is hardly addressed in Dutch vocational education. Qualification profiles include competencies such as “care for quality” but the appropriate statistical techniques are generally not specified or taught (e.g., competentie.kenteq.nl/cms/publish/content/ showpage.asp?pageid=700).

More generally there is a discrepancy between the competencies required at work and what is taught at school. The TmL project has shown that technical work (such as industrial production) is increasingly mediated by technology (e.g., via control panels). This implies that students should develop TmC rather than abstractly represented mathematical knowledge. Rather than calculating by hand or with a calculator, students should learn how to use the mathematics embedded in the tools and develop a situated model of the variables relevant in the work process (Bakker et al., 2006). The need for TmC is especially apparent in communication: within a work team, with managers or customers etc. We expect similar results for multimedia design and ICT system management, which are included in the present proposal. Note that the proposed project is not a replication of the TmL project because it takes place in Dutch vocational colleges and focuses on engineering, graphical design and ICT system management whereas the TmL project took place in English workplaces and focused on manufacturing (Bakker et al., 2006; Hoyles et al., in press) and financial services (Kent et al., in press).

### Hypothesis and research questions

The proposed project intends to test the following hypothesis:

- TmC that are tailored to the context and tools of professions will improve students’ ability to perform core tasks during at school and during work practice periods at learning-work sites (leerbedrijven).

The theoretical framework mentioned above and the Dutch situation of VET lead to the following three questions, answered in three research phases. To identify the TmC that need to be developed the main question in the first, ethnographic phase is:

- What Techno-mathematical Competencies are required in technical workplaces and can serve as instructional targets within the existing qualification profiles?

- The first research phase will yield authentic core tasks that are relevant for the professions at stake and have a clear techno-mathematical aspect.
- The second, design-based research phase will focus on the following main question, where core tasks can be performed at school or at learning-work sites:

- How can new learning arrangements support the development of the techno-mathematical competencies required for performing core tasks at school and learning-work sites?

- The second research phase will involve the design of learning arrangements for developing TmC in VET, collaboratively with teachers and practical work supervisors, and the analysis of participants’ learning processes. At college, students will work through new learning materials focusing on techno-mathematical knowledge relevant to practical core tasks, and they will reflect on their experiences during practice periods in relation to relevant knowledge. Computer tools are designed to represent the techno-mathematical aspects of those core tasks in such a way that they facilitate recontextualisation. The teaching experiments aim for discussion amongst students about school and work contexts in relation to the tools to support integration of different knowledge forms across the boundaries. Because mathematics education in VET does not have abstraction and generality as its central tenets, typical instructional design models known from mathematics education research that suggest a progression from situational to formal mathematical knowledge are not applicable. Hence students’ progression in recontextualising their techno-mathematical knowledge is to be judged not by an increasing level of mathematical generality but by the integration of appropriate contextual and techno-mathematical knowledge and skills in new problem situations and the success of dealing with these situations in performing core tasks as judged by practical supervisors. In the third, comparative phase the question is:

- In what ways does the development of TmC improve students’ performance of core tasks whether at college or learning-work sites?

- The research will be carried out in the mainly college-based BOL stream of senior secondary vocational education (mbo) rather than in the mainly work-based BBL stream because students in BOL spend 20-40% of their time at work rather than 60+% in BBL. It is therefore expected that the design of the learning environments is more feasible in BOL than in BBL.

## 14.2 setup and methods

The method involves a transition from ethnographic studies (to answer question 1) to design-based research (question 2) with a comparative last phase (question 3). The schools and workplaces fall under the technical sector (Techniek) of the PGO Consortium and Fontys, multimedia design of the Grafisch Lyceum Utrecht, and ICT system management of ROC Utrecht. In most colleges, mathematics is taught as a separate subject, but it is increasingly addressed in relation to projects. Collaboration with the TOP3C project (www.fontys.nl/top3c) ensures close contact with several companies (mechanical, electrical and process engineering) via Goris, who is a member of the advisory board. The Freudenthal Institute is one of the accredited learning-work sites for the other sectors.

### Phase 1: ethnographic studies and study of existing VET

Surveys generally do not yield the type of data required to identify areas in which TmC might be an issue because they yield data on a more aggregate level. Ethnographic studies will therefore be carried out in learning-work sites linked to mbo schools to answer question 1. We take a “theory-driven” approach (Pawson & Tilley, 1997) to study specific phenomena such as TmC, which are already known from previous research. Semi-structured interviews will be carried out with practical work supervisors and college teachers in the technical sector (electrical and mechanical engineering) and students will be interviewed during apprenticeships or practice periods about the techno-mathematical knowledge they feel they have not sufficiently developed. The data gathered (audio, pictures, calculations and copies of artefacts such as graphs) will be used to identify key elements of the activity systems in which they work: the tools they use (in particular the techno-mathematical ones), the community in which they work as well as techno-mathematical aspects of core tasks. The post-doctoral and junior researcher will analyse the data resources according to methods described in Hammersley and Atkinson (1995). It is expected that the TmC involve interpreting graphs of work process (in engineering sector), mathematical transformations and working with coordinates (in the multimedia design sector). At least twelve activity systems in four different companies will be analysed in total to define authentic core tasks. In collaboration with teachers and supervisors, a selection will be made of core tasks to address their techno-mathematical aspects more explicitly in relation to projects and practice periods.

### Phase 2: design-based research to enhance TmC

To answer question 2, the methodology of design research (e.g., Edelson, 2002) is appropriate because the project aims more at “understanding how” than at “knowing whether” learning arrangements can support the development of TmC in an ecologically valid way. These learning arrangements can be developed on the theoretical and practical knowledge basis of the TmL project and the design experience of the TWIN curriculum authors Goris and Van der Kooij, who are part of the advisory board. Design-based research as deployed here aims at shaping innovative instructional sequences, developing a local (domain-specific) instruction theory and general theoretical knowledge (progressive recontextualisation, boundary crossing). The design is based on design heuristics from VET research (as summarised by Fürstenau, 2003) and Realistic Mathematics Education theory (Gravemeijer, 1994) to ensure a sound theoretical design basis. The focus will be on the techno-mathematical side of core tasks (e.g., quality control, defining mathematical functions in ICT systems).

The research set-up is characterised by an iterative, cyclic design. Both phase 2 and 3 consist of a preliminary stage in which instructional activities are designed that embody task-specific conjectures, a teaching experiment stage in which the conjectures that form the basis of the student activities are tested, and a retrospective stage which generates revised conjectures (Gravemeijer & Cobb, 2006). The first preliminary stage is based on the findings of the ethnographic phase 1. The designed instructional sequence aims at developing TmC, in particular graphs of workplace data to make decisions based on the core tasks identified in research phase 1.

The three teaching experiments in each of the phases 2 and 3 involve at least 20 students and will take place at college (engineering) or during a practice period (graphical design and ICT system management). In phases 2 and 3, data resources are audio and video recordings of the students during the teaching experiment and log-files of their work (Camtasia) with the computer tools (Flash). The post-doctoral and junior researcher act as participating observers; observations and interventions are based on the pre-formulated conjectures of the local instruction theory (e.g., about students’ responses to an instructional activity and what they learn from it). With the help of software for data analysis (MEPA), the data resources will be used to test these conjectures.

The researcher will record decisions in a logbook to capture the empirical basis and theoretical considerations for choices made during the design process. He will also record examples of boundary crossing between college teachers and practical supervisors and other interested parties. For reach of the three contexts, three techno-mathematical core tasks will be designed collaboratively with teachers and supervisors to analyse students’ progressive recontextualisation (beginning, middle and end of the teaching experiment or practice period). Assessment takes place by the teacher or supervisor.

In the retrospective stage, the theoretical orientation towards activity theory, competencies literature and in particular the TmL research forms the interpretative and explanatory framework. In particular, examples of boundary crossing situations will be analysed to understand better how recontextualisation can be supported. The results of the analysis of the students’ learning include indicative conclusions on the task-specific conjectures and new insights that are embedded into the design of the instructional sequence in the next phase. They also include the development of a local instruction theory and suggestions for analysing progressive recontextualisation of techno-mathematical knowledge in phase 3.

To assist in the design of materials, interpretation of data and developing the theoretical frameworks, an advisory board of nine researchers and educators has been established: seven VET, mathematics and workplace researchers (3 Dutch, 4 UK: Jonker, Onstenk, Wijers, Brown, Guile, Hoyles and Noss) and two authors of the TWIN curriculum (Van der Kooij and Goris). The mbo teachers and practical work supervisors who are involved in the design process and teaching experiments will also be invited. During a two-day expert meeting theoretical themes that arise will be discussed and results from the international research teams on similar themes will be compared.

### Phase 3: the comparative phase

During the preliminary stage of this phase, the design team revises the instructional sequence on the basis of the results of phase 2 and the advisory board members individually reflect on the revised design. The style of working is similar to the one described for phase 2, but – to answer question 3 – this time a comparison will be made on the three core tasks identified for analysing progressive recontextualisation between the students of the experimental groups (at least 20 across three colleges) and at least 16 other students who will function as the control groups. These students do the same work projects or similar practice periods as the experimental students and their performance on the three core tasks is analysed but they are not involved in the teaching experiments that specifically aim at developing TmC. Practical work supervisors will assess their performance of core tasks. The teaching experiment in phase 3 includes a pre-test and a post-test on techno-mathematical knowledge. Students in both groups will be matched on their performance on the pre-test and on the first core task.

The method of analysis in phase 3 is similar to that of phase 2, but focuses on finding confirmations and refutations of the task-specific conjectures stated in the previous phase. Students’ work with the tools and their discussions are coded by both the post-doctoral and junior researcher and tested for interreliability. After that, a second two-day expert meeting is held with the advisory board to evaluate the analysis and the conclusions, to make international comparisons and to discuss theoretical themes such as boundary crossing, progressive recontextualisation and TmC in relation to the competencies issues. Then a revised local instruction theory is formulated and the empirical findings will be used to contribute – where appropriate jointly with members of the advisory board – to the development of the theoretical themes.

## References

- Bakker, A., Hoyles, C., Kent, P. and Noss, R. (2005). Designing Learning Opportunities for Techno-mathematical Literacies in Financial Workplaces: A status report. (Translator, Trans.). London: Institute of Education, University of London.
- Boundary object
- Kent, P., Hoyles, C., Noss, R. and Guile, D. (2004). Techno-mathematical Literacies in workplace activity.
- Techno-Mathematical Literacy
- Tuomi-Gröhn, T. and Engeström, Y. (2003). Conceptualizing transfer: From standard notions to developmental perspectives (In T. Tuomi-Gröhn and Y. Engeström (Eds.), Between school and work: New perspectives on transfer and boundary-crossing (pp. 19-38). Amsterdam: Pergamon.

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