1 Introduction
The current job market is a competitive one, with employers requiring work experience in addition to a degree, to support recruitment and selection processes (Tomlinson, 2008; Tomlinson et al., 2017). There are various models to achieve work experience while studying, such as work-based projects and paid-work placements of specific duration. A contrasting approach, degree apprenticeships (referred to as Graduate Apprenticeships in Scotland), means studying while working, whereby the graduate apprentice (GA) is simultaneously an employee and a student. While there are differences within the devolved nations of the UK, it is accepted that the GA system introduced (Smith et al., 2021b) is, in part, inspired by the German dual system (Protsch & Solga, 2016). For example, in the dual system, employers pay apprentices’ salaries, equipment costs and training, while government funds their study in specialist vocational schools. This approach has been criticized because the training places can be accessed, especially for IT specialists, depending on school attainment, which can be directly correlated to socio-economic background, and therefore is a selective scheme. Alternatively, the co-operative education model implemented in US and Canada (Angerilli, Johnston, & Gajdamaschko, 2003) offers intensive work experience but with no long-term employment, and generally targets school-leavers with little experience of the workplace environment (Linn, 2015).
Being enrolled in a Graduate Apprenticeship (GAP) programme offers an opportunity to employees working in diverse industries to gain a higher qualification without leaving employment. GAP may also be chosen, rather than a traditional degree, due to work experience integration and avoidance of student debt (Taylor-Smith et al., 2019b). Moreover, apprentices develop essential industry skills whilst maintaining the active workforce.
In Scotland, the process was administered until 2021 by Skills Development Scotland (SDS), an executive public body of the Scottish government. SDS brought employers together to develop the frameworks that shaped graduate apprenticeships in twelve fields, with the majority in STEM areas. Higher Education Institutes (HEI s) proposed implementation of GA degree programmes
Therefore, within the Scottish sector, GAP s are considered an initiative to create in-house degree-qualified employees with key skills customized to the demand of their industries. In the UK, work-based learning and degree apprenticeships have been widely exploited to address the perceived graduate skills gap (Fincher & Finlay, 2016), at the same time employers reported having difficulties in recruiting skilled employees (Shadbolt, 2016). As a result, employers experience several benefits from being part of degree apprenticeships and having apprentices such as:
- –GA s receive structured and supported ways of learning,
- –GA s gain new skill sets and a fresh perspective,
- –The organisation improves its quality of service,
- –GAP s help to retain employees,
- –GAP s help existing employees to enhance their skills.
As the apprentices are employed in diverse roles by various employers (from public sector to international companies to local start-ups), different experiences, opportunities and challenges are brought to their studies especially the work-based learning elements. The use of a work-based learning approach (Boud & Solomon, 2001) is considered as one of the benefits of the graduate apprenticeship that ensures reflection on practical applications in the business environment automatically.
As alluded to previously, the apprentice will attend university one day a week. Referred to as “release day”, GA s attend timetabled sessions and undertake self-directed and supported learning. One of the benefits of GAP s is the use of the work-based learning (WBL) approach that encourages drawing on experience of practical applications in the business environment. Work-based
Although the workplace is the obvious location for learning, the process of learning through and at work does not necessarily come naturally (Raelin, 2008). Organisations that adopt WBL approaches tend to value collaboration over individualism and have clarity regarding mission and goals. We can see, therefore, the potential for ontological thresholds to appear as GA s negotiate simultaneous and competing identities. Comparing GA s to other groups such as PhD students (Keefer, 2015; Wisker, 2016), and academics engaging with SoTL (Tierney, 2017), the ontological shifts experienced are similar as GA s negotiate dual identities of novice student and expert workplace practitioner. Issues such as negotiating a new paradigm, the “black box” of academic methodologies, and assumptions of background knowledge are all at play here, and should be negotiated with GA s. It is important for GAP teams to be open about academic expectations, whilst at the same time keeping communication open in order to encourage GA s to voice their uncertainties and anxieties.
Work-based learning programmes typically share the following six characteristics (Boud & Solomon, 2001):
- –Establishment of a partnership between an educational institution and an external organization, in order to foster learning, enable infrastructure, and support learning.
- –Learners involved are employees of the external organisation with different needs and these change over time. Therefore, a learning plan is essential for each learner.
- –Work-based activity is incorporated in the curriculum where the programme is informed by the needs of workplaces or learners.
- –The higher education institution assesses the learning outcomes of the WBL programmes with respect to a framework of levels and standards.
- –A flexible starting point is established when the learners engage in a process of recognition of prior experiential learning, which includes workplace experience in addition to certificates and courses.
- –
Learning projects undertaken at the workplace are a crucial part of WBL and apprentices tend to enjoy and engage with these projects more.
While GA programmes can be challenging, feedback gathered from apprentices related to work-based learning courses are positive, and include:
- –good use of practical applications during lectures,
- –great opportunity to learn the theory behind the practice while simultaneously gaining experience,
- –practical lectures which facilitate the application of theory.
However, one of our observations is that teaching core courses in a GA program is considered a challenge, not only because these courses are heavily centred around theory and concepts, but also due to the apprentices’ diversity in professional background, skills, qualifications, and experience (Faulkner-Jones et al., 2022). This makes core courses ripe for epistemological thresholds, as GA s come to terms with gaps in theoretical knowledge, their extensive practical experience, and the frustrations of trying to bring practice and theory together.
The GA programme at Heriot-Watt is designed to have two core courses per academic year which counts for 20% of university learning and the other 80% is WBL. The core courses have a similar structure of lectures and tutorials as the undergraduate courses in non-GA programmes, although the cohort is quite different. For our DS GA programme, the core courses are heavily based on mathematical and statistical skills which is a major element of our programme (Dziallas et al., 2021). We argue that these mathematical core courses potentially have numerous potential epistemological threshold concepts, as GA s encounter theoretical concepts which are novel to them, and necessary to progress in the field (Meyer & Land, 2003; Breen & O’Shea, 2016; Farjudian, 2020). The mastery of these concepts is necessary to make progress in the discipline which transforms the GA s’ expertise (Eckerdal et al., 2006). The features or characteristics of the threshold concepts that resonate strongly with this research (Eckerdal et al., 2006; Meyer & Land, 2003, 2005; Cousin, 2010) are:
- –Transformative: When the threshold concept involves a significant shift in the perception of a subject
- –Irreversible: Once a threshold concept is understood it is difficult for it to be unlearned.
- –Integrative: Bring together different aspects of the subject that previously were not apparent
- –Troublesome: Encountering difficulties with language, ritualised and conceptually difficult knowledge, or language that is troublesome
- –Bounded: Threshold concept that may lie on the border between conceptual spaces or marks the demarcation line between disciplinary areas (Breen &
O’Shea, 2016). For example, when limit definition is understood, this opens the door to the field of Analysis and sits on the boundary between Calculus and Analysis courses (Meyer & Land, 2003). - –Liminality: Threshold concepts may position students within a liminal space where their understanding is rendered unstable in the oscillation between old and new understanding (Pettersson, 2011, 2012).
In the coming sections, we define the challenges that the apprentices face and present some examples of threshold concepts, exploring characteristics which helped us identify them. We discuss the complexity of GA programme provision, and contemplate proposed solutions to the challenges we have uncovered.
2 The Challenge for GA s
The challenge for GA s within a programme is not learning subject material. For the most part, GA s have a wealth of practical experience which they are able to apply in a variety of situations. Their challenge is linking their practical expertise with theory, and understanding the different ways knowledge is presented. As an example, one of the mathematical core courses taught to 2nd year apprentices is Multivariate Calculus and Linear Algebra. Some of the mathematical topics included in the course such as differentiation, Taylor series and Eigenvalues and Eigenvectors could be seen as potential threshold concepts (Breen & O’Shea, 2016; Cousin, 2010; Loch & McLoughlin, 2012; Farjudian, 2020). These mathematical threshold concepts should be considered alongside the identity threshold concepts that GA s encounter, because of their unique backgrounds. In particular, the liminality of dual identity of professional and student should be sensitively handled when GA s struggle with theoretical mathematical concepts.
3 Mathematical Threshold Concepts
Differentiation is a fundamental concept in mathematics. It is often taught in an abstract way, and the failure to connect the operations of differentiation to its application can cause issues in understanding, leaving learners frustrated. However it is vital that learners understand differentiation in order to then comprehend further mathematical concepts. For example, understanding differentiation allows learners to then understand Taylor series as this is an application of differentiation. Understanding differentiation also widens boundaries of knowledge and addresses areas of conceptually difficult
Taylor series is a topic included in the HWU GA core course that relies heavily on differentiation (Worsley et al., 2008). This is an example of a threshold concept with a transformative feature. Moreover, because Taylor series of a function is defined as an infinite sum of terms that are expressed in terms of the function’s derivatives at a single point. This definition is a rich source of alien knowledge for most of the students and even some students who know the definition do not want to understand what it means, as they fear this will confuse them even more. The feelings of liminality associated with understanding Taylor series points to a troublesome threshold concept. When designing our lectures to explain this concept our objective was to find interactive and engaging ways to explain and visualise the concept to help the students to relate the concept to its wider applications.
Eigenvalues and Eigenvectors are normally troublesome for students as they are conceptually difficult. Moreover, these concepts are also integrative as their mastery opens up new areas of mathematics meaning that it is essential that GA s understand linear algebra and the application of matrices and vectors to be able to move forward.
Vectors and matrices are the building blocks for Eigenvalues and Eigenvectors (Loch & McLoughlin, 2012; Breen & O’Shea, 2016). As with the previously described Taylor series, vectors and matrices are another transformative and troublesome example. This may be because GA s do not use them in their practice, but also points to the GA s lack of theory and mathematical rules compared to their practical expertise.
The core course in the GAP is crucial, as the apprentices are being prepared to complete other courses like Machine Learning and Optimization, which represent vital elements of the Data Science programme. For example, Machine Learning is a combination of mathematics and computer science. Hence, one can observe that the whole course with its mathematical concepts can be seen with some potential threshold concepts with transformation, integration and reconstitution. If the GA s are well prepared to understand and visualise these concepts, then these concepts are also irreversible. Moreover, if they learn to apply it at their work, they will continue to exploit these concepts and their applications.
4 Proposed Solution and Findings
In the previous sections we have identified the potential for threshold concepts which GA s may encounter, both mathematical and the Gas’ dual identity. In order to mitigate these, and to support GA s in their academic studies,
Our GA population tackled numerous programming examples in order to master complex mathematical problems. For example, in Figure 28.1, the Python code for the plotting of the Taylor series Approximation of $\cos(x)$ is provided to the apprentices. Then, using this code they can make some changes to solve similar problems numerically.
An example to illustrate the visualisation of the plot of Taylor series approximation of cos (x) up to the 15th degree
Another example of how programming can help GA s understand complex mathematical concepts is plotting the stationary points of the given function. When we ask GA s to do this, the local minima, local maxima and saddle points can be observed as in Figure 28.2. Plotting the function and its relevant stationary points assists the learning and understanding of the GA s, especially if they are encountering it for the first time. This dual approach also supports long term recall and problem solving as GA s can apply these solutions in different contexts.
Plotting the stationary points on 3D function showing local maxima, minima and saddle points
Using programming to support solving complex problems, encourages GA s to apply and integrate their knowledge. For example, solving a Taylor series problem as shown in Figure 28.3, the apprentices are asked to learn about relative error percentage by calculating the cubic root of a number and comparing the solution with the approximation using Taylor series. These examples and
Application example to Taylor series approximation
Finally, to enhance their learning, reflection on the obtained results was required from GA s. They were asked to identify the differences between the numerical and analytical solutions (Figure 28.4). For example, in matrix diagonalisation, they are asked to check if this is possible or not by calculating Eigenvalues and Eigenvectors. Hence, solving the problem analytically and then confirming their results by creating a code to solve the problem numerically. In the reflection they would notice there is a difference, and the reason is the Eigenvectors computed numerically are normalised. Moreover, multiplying the Eigenvector matrices with the diagonal matrix of Eigenvalues to obtain the initial matrix.
Example of reflection on obtained results
To conclude, we use different software programming tools such as Matlab, Python etc. when teaching mathematical core courses (Howe et al., 1982; Abdulwahed et al., 2012) in order to support GA learning. We monitor learning through GA performance in tutorials and assessment submissions. We found that, as a result of the dual approach, apprentices are more confident with their results and can visualise and better understand the potential threshold concepts discussed before. This approach also offers the apprentices with an interactive environment to explore programming based procedural and functional approaches to problem-solving. Moreover, our apprentices have been asked to validate their analytical results with the numerical ones. This assures
5 Reflection and Future Work
Although the proposed approach was a success from the instructor point of view and based on the GA s meeting the learning outcomes of the course, there were some challenges, first to design the exercises and examples in lectures, tutorials and assessments and trying a new approach that the apprentices have not experienced before (Brodie & Irving, 2007). Furthermore, the GA s’ fear of programming especially if it is not their strongest skill and this leads to high rates of retention in STEM subjects (Sithole et al., 2017). Since our objective was not to assess the apprentices’ programming skills, we would provide them with the codes and allow the apprentices to make changes to the code based on the given example. In some examples and exercises the coding was optional, and then we would struggle to encourage the apprentice to submit these optional questions to enhance their learning. Thus, for future work, we plan to include less optional coding exercises, and provide more resources and hints to help the apprentices solve the questions numerically.
To use this approach for more advanced mathematical courses, the GA s will be asked to complete small and manageable tasks to reach their goals
Finally, although this approach was chosen for the graduate apprenticeships due to its unique curriculum design and the apprentices who rely heavily on merging working with studying, for our future work, this approach would be investigated for the undergraduate programmes (non-GA courses) to enhance the integration between the theoretical and practical aspects. This allows the students to grasp these concepts and enhance their practical skills to be ready for their real-world applications.
References
Abdulwahed, M., Jaworski, B., & Crawford, A. (2012). Innovative approaches to teaching mathematics in higher education: A review and critique. Nationellt centrum för matematikutbildning. Figshare. https://hdl.handle.net/2134/11988
Angerilli, N., Johnston, N., & Gajdamaschko, N. (2003). Measuring the co-op experience: Assessing student and practitioner perceptions of learning and transfer using Q-Methodology. In P. Lynn, A. Howard, & E. Miller (Eds.), Handbook on research in co-operative education and internships (pp. 157–191). Lawrence Erlbaum Associates, Inc.
Bergin, S., & Reilly, R. (2005). Programming: Factors that influence success. In Proceedings of the 36th SIGCSE technical symposium on Computer science education (pp. 411–415). https://doi.org/10.1145/1047344.1047480
Boud, D., & Solomon, N. (2001). Work-based learning: A new higher education? McGraw-Hill Education.
Breen, S., & O’Shea, A. (2016). Threshold concepts and undergraduate mathematics teaching. PRIMUS, 26(9), 837–847. https://doi.org/10.1080/10511970.2016.1191573
Brodie, P., & Irving, K. (2007). Assessment in work-based learning: Investigating a pedagogical approach to enhance student learning. Assessment & Evaluation in Higher Education, 32(1), 11–19. https://doi.org/10.1080/02602930600848218
Cousin, G. (2010). Neither teacher-centred nor student-centred: Threshold concepts and research partnerships. Journal of Learning Development in Higher Education, 2, 1–9.
Dziallas, S., Fincher, S., Barr, M., & Cutts, Q. (2021). Learning in context: A first look at a graduate apprenticeship. 21st Koli Calling International Conference on Computing Education Research (Koli Calling ‘21). Association for Computing Machinery Article 19, 1–11. https://doi.org/10.1145/3488042.3490020
Eckerdal, A., McCartney, R., Moström, J.E., Ratcliffe, M., Sanders, K., & Zander, C. (2006). Putting threshold concepts into context in computer science education. ACM Sigcse Bulletin, 38(3), 103–107. https://doi.org/10.1145/1140123.1140154
Farjudian, A. (2022). Bridging mathematics and computer science through threshold concepts. IEEE Transactions on Education, 1–7. https://doi.org/10.1109/TE.2022.3200162
Faulkner-Jones, A. J., du Plessis Love, O., Ilyas, A., Zahid, A., & Westacott, R. (2022). Deep learning: A case for graduate apprenticeships. In Proceedings of the 8th International Symposium for Engineering Education. University of Strathclyde, Glasgow.
Fernández, M., Wegerif, R., Mercer, N., & Rojas-Drummond, S. (2002). Re-conceptualizing “scaffolding” and the zone of proximal development in the context of symmetrical collaborative learning. The Journal of Classroom Interaction, 36(2/1), 40–54.
Fincher, S., & Finlay, J. (2016). Computing graduate employability: Sharing practice. In Project Report. Council of Profesors and Heads of Computing. Kent, UK. https://kar.kent.ac.uk/53848/
Howe, J. A., Ross, P. M., Johnson, K. R., Plane, F., & Inglis, R. (1982). Teaching mathematics through programming in the classroom. Computers & Education, 6(1), 85–91. https://doi.org/10.1016/0360-1315(82)90016-1
Juan, A. A., Steegmann, C., Huertas, A., Jesus Martinez, M., & Simosa, J. (2011). Teaching mathematics online in the European Area of Higher Education: An instructor’s point of view. International Journal of Mathematical Education in Science and Technology, 42(2), 141–153. https://doi.org/10.1080/0020739X.2010.526254
Kurt, S. (2020). Vygotsky’s zone of proximal development and scaffolding. https://educationaltechnology.net/vygotskys-zone-of-proximal-development-and-scaffolding/
Keefer, J. M. (2015). Experiencing doctoral liminality as a conceptual threshold and how supervisors can use it. Innovations in Education and Teaching International, 52(1), 17–28. https://doi.org/10.1080/14703297.2014.981839
Lemanski, T., Mewis, R., & Overton, T. (2011). An introduction to work-based learning. UK, Physical Sciences Centre, Department of Chemistry, University of Hull: Higher Education Academy UK Physical Sciences Centre.
Linn, P. L. (2015). A Lifespan perspective on cooperative education learning: A grounded theory. Asia-Pacific Journal of Cooperative Education, 16(4), 301–326.
Loch, B., & McLoughlin, C. (2012). Teaching threshold concepts in engineering mathematics using MathsCasts. In L. Mann & S. Daniel (Eds.), Proceedings of the Australasian Association for Engineering Education Conference (23rd). The Engineering & Science Education Research (ESER) group, Faculty of Engineering & Industrial Sciences, Swinburne University of Technology, Melbourne, Australia.
Meyer, J. H. F., & Land, R. (2003). Threshold concepts and troublesome knowledge: Linkages to ways of thinking and practising within the disciplines. In C. Rust (Ed.), Improving student learning theory and practice – 10 years on (pp. 412–424). Oxford Centre for Staff and Learning Development.
Meyer, J. H. F., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): Epistemological considerations and a conceptual framework for teaching and learning. Higher Education, 49, 373–388. https://doi.org/10.1007/s10734-004-6779-5
Oates, G. N., Reaburn, R. L., Brideson, M. A., & Dharmadasa, H. K. (2018). Understanding of limits and differentiation as threshold concepts in a first-year mathematics course. The Eleventh Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics, 108–120.
Pettersson, K. (2011). Threshold concepts: A framework for research in university mathematics education. Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education, 2063–2072.
Pettersson, K. (2012). The threshold concept of a function–A case study of a student’s development of her understanding. Eighth Swedish Mathematics Education Research Seminar, Umeå, Sweden.
Powell, A. (2019). Apprenticeships and skills policy in England. House of Commons Briefing Paper CBP 03052. https://researchbriefings.files.parliament.uk/documents/SN03052/SN03052.pdf
Protsch, P., & Solga, H. (2016). The Social stratification of the German VET system. Journal of Education and Work, 29(6), 637–661. https://doi.org/10.1080/13639080.2015.1024643
Raelin, J. A. (2008). Work-based learning: Bridging knowledge and action in the workplace. John Wiley & Sons.
Ryan, A. M., Gass, G., Gass, S., Ryan, K., & Van Dommelen, J. (2014, July 9–11). Quantitative reasoning and visualizing data across the sciences. Fifth International Biennial Threshold Concepts in Practice, Durham University, Durham, UK.
Shadbolt, N. (2016). Shadbolt review of computer science degree accreditation and graduate employability. UK Government Report. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/518575/ind-16-5-shadbolt-review-computer-science-graduate-employability.pdf
SDS (Skills Development Scotland). (2016). Graduate level apprenticeships: Pathfinder delivery. Competitive Funding Offer Ref No. G16-026. Skills Development Scotland.
Sithole, A., Chiyaka, E. T., McCarthy, P., Mupinga, D. M., Bucklein, B. K., & Kibirige, J. (2017). Student attraction, persistence and retention in STEM programs: Successes and continuing challenges. Higher Education Studies, 7(1), 46–59. https://doi.org/10.5539/hes.v7n1p46
Smith, S., Caddell, M., Taylor-Smith, E., Smith C., & Varey, T. (2021a). Degree apprenticeships – a win-win model? A comparison of policy aims with the expectations and experiences of apprentices. Journal of Vocational Education & Training, 73(4), 505–525. https://doi.org/10.1080/13636820.2020.1744690
Smith, S., Taylor-Smith, E., Fabian, K., Zarb, M., Paterson, J., Barr, M. and Berg, T. (2021b). A multi-institutional exploration of the social mobility potential of degree apprenticeships, Journal of Education and Work, 34(4), 488–503. https://doi.org/10.1080/13639080.2021.1946494
Taylor-Smith, E., Smith, S., & Smith, C. (2019a). Identity and belonging for graduate apprenticeships in computing: The experience of first cohort degree apprentices in Scotland. In Proceedings of the 2019 ACM Conference on Innovation and Technology in Computer Science Education, 2–8. https://doi.org/10.1145/3304221.3319753
Taylor-Smith, E., Smith, S., Fabian, K., Berg, T., Meharg, D., & Varey, A. (2019b). Bridging the digital skills gap: Are computing degree apprenticeships the answer? Proceedings of the 2019 ACM Conference on Innovation and Technology in Computer Science Education, 126–132. https://doi.org/10.1145/3304221.3319744
Tierney, A. M. (2017). Threshold concepts in academic practice: Engagement with the scholarship of teaching and learning. Practice and Evidence of the Scholarship of Teaching and Learning in Higher Education, 12(2), 165–184.
Tomlinson, M. (2008). ‘The Degree Is Not Enough’: Students’ perceptions of the role of higher education credentials for graduate work and employability. British Journal of Sociology of Education, 29(1), 49–61. https://doi.org/10.1080/01425690701737457
Tomlinson, M., McCafferty, H., Fuge, H., & Wood, K. (2017). Resources and Readiness: The graduate capital perspective as a new approach to graduate employability. Journal of the National Institute for Career Education and Counselling, 38(1), 28–35. https://doi.org/10.20856/jnicec.3805
Wisker, G. (2016). Beyond blockages to ownership, agency and articulation: Liminal spaces and conceptual threshold crossing in doctoral learning. In R. Land, J. H. F. Meyer, & M. T. Flanagan (Eds.), Threshold concepts in practice (pp. 165–178). Sense. https://doi.org/10.1007/978-94-6300-512-8_13
Worsley, S., Bulmer, M., & O’Brien, M. (2008). Threshold concepts and troublesome knowledge in a second level mathematics course. Proceedings of the Visualisation for Concept Development Symposium.