The kernel of the dual system of vocational education is formed by its two pillars, learning at vocational schools and learning in enterprises. Their relationship, however, widely discussed in earlier times, has re-appeared on the agenda within the frame of connectivity theory (as originally proposed by Griffith and Guile 2001, 2003, later elaborated by Tynjälä and Stenström 2009), and by the theory of complementarity of vocational education (first by Jongebloed 1998, 2004).

Although the original authors of both theories reject the role of learning transfer as some sort of missing link between theory and practice (Griffith & Guile 2003, 61, Jongebloed 2004, 32), they put back the spotlight on this question. In fact, we expect to find a vast potential in explaining the gap between theoretical and practical learning by making use of the theories of learning transfer (as put forward by Thorndike 1923, Osgood 1948, Judd 1908, Overing & Travers 1966, Harlow 1949, Katona 1940, and many others). This leads to interpret the process of learning of theory at school, and the process of practise at the workplace as two a priori independent processes of learning. As such, any kind of theoretical learning can only be explained by its factual value, generated by the positive influence it expectedly exercises on the process of learning in the workplace. Would theory learning not produce anything like support, enhancement, or, improvement of the following learning process in the real world, then nobody would be ready to spend any time on the learning of any theory.

The two learning processes apparently fulfil the defining criteria of positive learning transfer. Therefore we should consider whether the difference between the two learning processes (the so-called transfer gap) can be interpreted as a distance, and be formally modelled. It also entails the question of how large the distance between learning processes technically can be – is there any extremal distance of learning transfer? After all, this is an eminently didactical issue, which asks for the development of measurement models to the learning distance in the hope of a better handling of this issue in teaching and learning. On this purpose, we will propose and discuss in our paper different modelling approaches to measuring the distance in learning transfer in dual systems of vocational education.

As a first step, we will introduce an integrated model of learning transfer which basically takes up the fundamental argument of Thorndike, that the probability of a successful learning transfer is the higher, the greater the number of “identical elements” between the learning processes L1 and L2 (Thorndike 1923, 359, see also an elaboration by Osgood 1949; the highest probability of success given by 1 non-identical element between L1 and L2). The other extreme, no identical element, does not leave any possibility that learning transfer may occur. This is to be understood in a generalised, not only in a behavioural way. This enlarged approach would also incorporate conditional (enhancing or disturbing) factors.

The simplest approach to these observations lies in modelling the width of the transfer gap by the Euclidian measure of distance, based on the quantity of identical elements. We also will discuss the approaches of Klauer (1989), and Gessler (2012). In particular Gessler’s model has got the merit to take into consideration both directions of learning transfer ( practice/ theory, too).  By introducing a new approach that will be based on graph theory, we define the measurement of the transfer gap as the measurement of the distance between two graphs. Furthermore, we will provide formal proof to its fitness.

Use of mathematical modelling (graph theory) on the base of research on learning transfer theory,

Enhancement of the previous models: Euclidian distance, for example, is insatisfactory, it might just be used as a quantitative proxy of similarity, but nothing more. It would be short of prognostic validity, whilst the calculus of any measure of similarity is aiming at the prognostics of the (probable) success of learning transfer by similarity or difference of the learning sets, just as postulated by Thorndike over a century ago.

Gessler, M. (2012). Lerntransfer in der beruflichen Weiterbildung – empirische Prüfung eines integrierten Rahmenmodells mittels Strukturgleichungsmodellierung. In: ZBW 108 (3), 362-393. Griffiths, T./ Guile, D. (2003). A Connective Model of Learning: the implications for work process knowledge. In: European Educational Research Journal 2 (2003) 1, http://eprints.ioe.ac.uk/3957/1/Guile%26Griffiths2001Learning113.pdf. Guile, D. / Griffiths, T. (2001). Learning through work experience. In: Journal of Education and work 14 (2001) 1, 113-131. Harlow, H. F. (1949). The Formation of Learning Sets, in: Psychological Review 56 (1), 51-65. Hendrickson, G. & Schroeder W. H. (1941). Transfer of Training in learning to Hit a Submerged Target, in: Journal of Educational Psychology 32 (3), 205-213. Jongebloed, H.-C. (1998). Komplementarität als Verhältnis: Lernen in dualer Struktur. In: Jongebloed, H.-C. (ed.): Wirtschaftspädagogik als Wissenschaft und Praxis- oder: Auf dem Wege zur Komplementarität als Prinzip (S. 259-286). Kiel. Jongebloed, H.-C. (2004). „Komplementarität“ als Prinzip beruflicher Bildung – oder Warum der „Lernfeldansatz“ weder dem Grunde nach funktionieren noch seine eigenen Ziele erreichen kann. www.uni-kiel.de/paedagogik/Jongebloed/publikationen/Jongebloed/Jongebloed_(2004)_Komplementaritaet.pdf. Judd, Charles H. (1908). The relation of special training to general intelligence, in: Educational Review 36, 28-42. Katona, George (1940). Organizing and memorizing. Studies in the Psychology of Learning and Teaching, New York. Klauer, K. J. (1989). Die Messung von Transferdistanzen. Ein Verfahren zur Bestimmung der Unähnlichkeit von Aufgabenanforderungen. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie 21 (2), 146-166. Osgood, Charles E. (1949). The Similarity Paradox in Human learning: A resolution, in: Psychological Review 56 (2), 132-143. Overing, Robert L.R. & Robert M. W. Travers (1966). Effect upon transfer of variations in training conditions, in: Journal of Educational Psychology 57, 179–188. Stenström, M.-L. (2009). Connecting Work and Learning Through Demonstrations of Vocational Skills – Experiences from the Finnish VET. In: Stenström, M.-L. / Tynjälä, P. (Hg.): Towards Integration of Work and Learning. Strategies for Connectivity and Transformation (221-238). Heidelberg. Thorndike, Edward L. (1923). The Psychology of Learning. Educational Psychology Vol. II, New York: Columbia University. Tynjälä, P. (2009). Connectivity and Transformation in Work-Related Learning – Theoretical Foundations. In Stenström, M.-L. / Tynjälä, P. (Hg.): Towards Integration of Work and Learning. Strategies for Connectivity and Transformation (11-37). Heidelberg.