Extreme Apprenticeship – Engaging undergraduate students on a mathematics course

Johanna Rämö 1 * , Thomas Vikberg 1
More Detail
1 Department of Mathematics and Statistics, University of Helsinki, Finland
* Corresponding Author
EUROPEAN J SCI MATH ED, Volume 2, Issue 2A, pp. 26-33. https://doi.org/10.30935/scimath/9623
OPEN ACCESS   421 Views   319 Downloads
Download Full Text (PDF)


This paper describes how an educational method called Extreme Apprenticeship has been used in teaching mathematics to undergraduates. The aim has been to facilitate the transition from secondary to tertiary education and to teach students the kind of skills they need in further studies and professional life. We report how the Extreme Apprenticeship method has been implemented in the course Linear algebra and matrices I at the University of Helsinki. This first year mathematics course has approximately 400 students each year. We compare the method with the traditional lecture-based approach that was in use before introducing the Extreme Apprenticeship method. We focus on how these two methods engage students. The results show that the Extreme Apprenticeship method managed to engage the students better than traditional teaching, as more students completed their coursework than before. Even though the new teaching method demanded a lot of personal effort from the students, they did not think that the workload was too big, and were pleased with the course.


Rämö, J., & Vikberg, T. (2014). Extreme Apprenticeship – Engaging undergraduate students on a mathematics course. European Journal of Science and Mathematics Education, 2(2A), 26-33. https://doi.org/10.30935/scimath/9623


  • Beck, K. (1999). Extreme Programming Explained: Embrace Change. XP Series. Addison-Wesley.
  • Biggs, J. and Tang, C. (2007). Teaching for quality learning at university: what the student does. Society for Research into Highter Education. McGraw-Hill.
  • Brown, J., Collins, A., and Duguid, P. (1989). Situated cognition culture of learning. Educ. Researcher, 18(1), 32.
  • Clark, M. and Lovric, M. (2008). Suggestion for a theoretical model for secondary-tertiary transition in mathematics. Mathematics Education Research Journal, 20, 25–37.
  • Collins, A., Brown, J., and Holum, A. (1991). Cognitive apprenticeship: Making thinking visible. American Educator, 15(3), 6–46.
  • Collins, A. and Greeno, J. G. (2010). Situative view of learning. In Aukrust, V. G., editor, Learning and Cognition (pp. 64–68). Elsevier Science.
  • Ericsson, K. A., Krampe, R. T., and Tesch-romer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review (pp. 363–406).
  • Greeno, J. G. (1997). On claims that answer the wrong questions. Educational Researcher, 26(1), 5–17.
  • Hautala, T., Romu, T., Rämö, J., and Vikberg, T. (2012). Extreme apprenticeship method in teaching university-level mathematics. Proceedings of the 12th International Congress on Mathematical Education: July 8-15, 2012, Seoul, Korea. International Commission on Mathematical Instruction (ICMI).
  • Keijonen, H., Kurhila, J., and Vihavainen, A. (2013). Carry-on effect of extreme apprenticeship. Proceedings of the Frontiers in Education Conference, 149–164.
  • Kurhila, J. and Vihavainen, A. (2011). Management, structures and tools to scale up personal advising in large programming courses. Proceedings of the 2011 conference on Information technology education, SIGITE ’11, 3–8. ACM.
  • Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture, and Activity, 3(3), 149–164.
  • Lave, J. and Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Learning in Doing. Cambridge University Press.
  • Lindblom-Ylänne, S., Mikkonen, J., Heikkilä, A., Parpala, A., and Pyhältö, K. (2009). Oppiminen yliopistossa. In Lindblom-Ylänne, S. and Nevgi, A., editors, Yliopisto-opettajan käsikirja (pp. 70–99). WSOYpro Oy.
  • Selden, A. and Selden, J. (2012). A belief affecting students’ success in problem solving and proving. Proceedings of the 12th International Congress on Mathematical Education. International Commission on Mathematical Instruction (ICMI).
  • Vihavainen, A., Paksula, M., and Luukkainen, M. (2011a). Extreme apprenticeship method in teaching programming for beginners. Proceedings of the 42nd ACM technical symposium on Computer science education, SIGCSE ’11, 93–98.
  • Vihavainen, A., Paksula, M., Luukkainen, M., and Kurhila, J. (2011b). Extreme apprenticeship method: key practices and upward scalability. Proceedings of the 16th annual joint conference on Innovation and technology in computer science education, ITiCSE ’11, 273–277. ACM.
  • Vygotsky, L. (1978). Mind in society: the development of higher psychological processes. Harvard University Press.
  • Wenger, E. (1998). Communities of Practice: Learning, Meaning, and Identity. Learning in Doing Series. Cambridge University Press.
  • Wood, D., Bruner, J. S., and Ross, G. (1976). The role of tutoring in problem solving. The Journal of Child Psychology and Psychiatry and Allied Disciplines, 17(2), 89–100.
  • Wood, L., Petocz, P., and Reid, A. (2012). Becoming a Mathematician: An International Perspective. Mathematics education library. Springer Netherlands.