Teaching Methodology



Research background

The products offered by Four Ferries are the result of more than 20 years of research in mathematics education at Abo Akademi University and University of Turku. The structured derivation format, on which many of the company products are bsed, was originally developed by Ralph-Johan Back and Joakim von Wright in the late 1990s, based on earlier ides by Edsger W. Dijkstra, Wim Feijen and Nettie van Gasteren on calculational proofs in the 1980s.

The method was further developed at the Learning and Reasoning Laboratory, a joint research lab for the IT-departments of Abo Akademi University and University of Turku. Funding for this research has been provided by the Academy of Finland, TEKES, EU, the Technology industry foundation of Finland and the Swedish Cultural Foundation in Finland. The structured derivation approach has been piloted in a large number of case studies at high school and junior high school level.

The first version of 4f Studio was  developed in the E-math project (2011- 2013). This was an EU project in the Central Baltic program (Interreg IV). The project created a digital learning environment for mathematics education at high school level, a collection of digital textbooks based on structured derivations, and piloted the textbooks and the environment in 15 different high schools in Finland, Sweden and Estonia.

The Swedish Cultural Foundation financed two different continuations of the EU project for 2014 - 2016. The first project (eMath 2) created a complete set of textbooks for high school mathematics (grades 10 - 12) according to Finnish national curriculum (in Finnish, Swedish and English). The second project, Virum, has generalized and extended the original E-math learning platform so that it now can be used for all school topics and at all school levels.

The company has continued developing the products after the academic projects ended. The support from the local industry development center (ELY) 2017-18 allowed us to generalise the Virum platform to a general purpose digital learning environment, 4f Studio. The support from the national industry development center (Business Finland 2018-19) allowed us to considerably extend the functionality and efficency of the automatic checker, so that it now supports all middle school and most advanced level high school math courses.

Research Projects

  • 1998 - 2004 Basic research on structured derivations (Ralph-Johan Back and Joakim von Wright). A number of projects funded by the Academy of Finland
  • 2000 - present. Learning and Reasoning laboratory was founded, focusing on Mathematics and Programming education. The laboratory is jointly managed by Abo Akademi University (Ralph-Johan Back) and University of Turku (Tapio Salakoski)
  • 2002 - 2007 Formal Methods in Programing (Ralph-Johan Back). A Center of Excellence in Research at the Academy of Finland.
  • 2002 - 2006 Academy Professorship Ralph-Johan Back was appointed Academy Professor by the Academy of Finland.
  • 2007 - 2014. Teachers training and tutorials on structured derivations. Supported by The Centennial Foundation of the Finnish Technology Industry.
  • 2008 - 2015. Teachers training courses on structured derivations and digital mathematics environments. Financed by the Finnish National Board of Education.
  • 2009 - 2011. Structured derivations in junior high school. Financed by the Finnish Technology Development Center (TEKES).
  • 2011 - 2013. E-math project. Building a digital platform for teaching mathematics at high school level. EU Central Baltic project, with participants from Finland (Turku, Aland), Sweden (Stockholm) and Estonia (Tallinn).
  • 2014 - 2016. eMath 2 project. Creating a complete textbook series for high school mathematics. Funded by the Swedish Cultural Foundation of Finland.
  • 2014 - 2017. ViRum project. Purpose is to create a computer based platform for digital courses over the net, for all topics in high school. Funded by the Swedish Cultural Foundation of Finland.
  • 2017 - 2018. 4f Studio project. Generlise the Virum environment to a general platform for digital courses. Funded by ELY-keskus.
  • 2018 - 2019. Automatic checking project. Extend automatic checking to cover the standard advanced level high school courses in mathmatics. Funded by Business Finland.


  • Teaching Mathematics in the Digital Age using Structured Derivations. (Back, R-J., Four Ferries Publishing, Amazon 2016)
  • Structured Derivations: Teaching Mathematical Reasoning in High school (Back, R-J., Four Ferries Publishing, Amazon 2015)
  • Teachers' Experience from Using Interactive E-books in the Classroom (Hägerstedt, E., Mannila, L., Salakoski, T. and Back, R-J., The Seventh Nordic Conference on Mathematics Education, NORMA14. Turku, Finland, June 2014)
  • Haasteet ja mahdollisuudet tietokonetuetussa matematiikan opetuksessa (Sallasmaa, P., Mannila, L., Peltomäki, M., Salakoski, T., Salmela, P. and Back, R-J, OPTEK ensitulosjulkaisu, 2011)
  • "It Takes Me Longer, but I Understand Better" – Student Feedback on Structured Derivations (Back, R-J., Mannila, L. and Wallin, S., International Journal of Mathematical Education in Science and Technology, Vol. 41, No. 5, pp. 575-593)
  • Structured Derivations: a Unified Proof Style for Teaching Mathematics. (Back R-J., Formal Aspects of Computing, 22(5), pp. 629-661)
  • Strukturerade härledningar i matematikundervisningen (Mannila, L. and Back, R-J. , Arkhimedes, No. 6, pp. 24-34)
  • Strukturerade härledningar ökar förståelsen (Mannila, L., Nämnaren, No. 3, pp. 18-25)
  • Writing Mathematics on the Computer Starting at Upper-Secondary Level (Peltomäki, M. and Mannila, L., ICL2010: 13th International Conference on Interactive Computer Aided Learning. Hasselt, Belgium. September 2010)
  • Student Justifications in High School Mathematics (Mannila, L., Wallin, S. and Back, R-J., CERME 6: Sixth Conference of European Research in Mathematics Education. Lyon, France, January-February, 2009)
  • An Empirical Evaluation of Structured Derivations in High School Mathematics (Peltomäki, M., and Back, R-J., ICMI 19: 19th ICMI Study Conference on Proof and Proving in Mathematics Education. Taipei, Taiwan, May 2009)
  • Promoting Students’ Justification Skills Using Structured Derivations (Mannila, L. and Wallin, S., ICMI 19: 19th ICMI Study Conference on Proof and Proving in Mathematics Education. Taipei, Taiwan, May 2009)
  • Rakenteiset päättelyketjut matematiikan opetuksessa (Peltomäki, M., Dimensio, No. 5, pp. 20-22)
  • Teaching Mathematics and Programming – New Approaches with Empirical Evaluation (Mannila, L. Phd Thesis, November 2009)
  • Structured Derivations: a Logic Based Approach to Teaching Mathematics (Back, R-J., Mannila, L., Peltomäki, M. and Sibelius, P., FORMED08, Budapest, March 2008)
  • Refinement Calculus: A Systematic Introduction. Chapters introducing the format (PDF): 3, 4, 6, 7 (Book by Back and von Wright, 1998)
  • Structured Calculation Proof (Back R-J., Grundy, J. and von Wright, J., 1997)
  • Structured Derivations: a Method for Doing High-School Mathematics Carefully (Back, R-J. and von Wright, J, 1999)
  • Field Tests of the Structured Derivations Method (Back, R-J., Sjöberg, M. and von Wright, J., 2002)