Rearranging equations: (concepts – misconceptions) × peer discussion

Maryna Lishchynska, Catherine Palmer, Julie Crowley


Transposition of formulae (also known as rearranging equations and changing the subject) is a skill vital for professionals in many fields of science and engineering. It is however a topic with which many students, and particularly students of weaker algebraic competency, struggle and often do not master sufficiently. This paper proposes an intervention strategy for improved teaching and learning of transposition of formulae at third level. The intervention aims to address three key issues thought to inhibit students’ understanding of the topic: (1) a lack of conceptual understanding of equations and equality, (2) prior misconceptions and (3) a fast paced learning environment that does not account for diversity in knowledge and aptitude. The strategy consists of three hour-long lesson plans that emphasise conceptual understanding while also dispelling the relevant misconceptions, using a peer discussion teaching model as a vehicle for consolidating and propagating the right concepts. To account for diversity in prior knowledge and aptitude an online tutorial on the topic of transposition has been developed using an online e-assessment platform that allows students to practice at their own pace and receive instant feedback as they progress.


transposition of formulae; rearranging equations; changing the subject; algebra

Full Text:



Barbieri, C. and Booth, J. L., 2016. Support for struggling students in algebra: Contributions of incorrect worked examples. Learning and Individual Differences, Volume 48, pp.36-44.

Barton, C., 2018. How I wish I'd taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.

Bloom, B., 1968. Learning for mastery. UCLA - CSEIP - Evaluation Comment. Available at: [Accessed 10 January 2020].

Bush, S. and Karp, K., 2013. Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3), pp.613-632.

Byrd, C. E., McNeil, N. M., Chesney, D. L. and Matthews, P. G., 2015. A specific misconception of the equal sign acts as a barrier to children's learning of early algebra. Learning and Individual Differences, Volume 38, pp.61-67.

Chow, T-C. F., 2011. Students' difficulties, misconceptions and attitudes towards learning Algebra: an intervention study to improve teaching and learning. Available at: [Accessed 10 January 2020].

Crouch, C., Watkins, J., Fagen, A. P., and Mazur, E., 2007. Peer Instruction: Engaging students one-on-one, all at once. Available at: [Accessed 10 January 2020].

Cummings, K. and Roberts, S. G., 2008. A study of peer instruction methods with high school physics students. AIP Conference Proceedings. Physics Education Research Conference, 1064(1), pp.103-106.

Fosnot, C., 2018. Conceptual Understanding. Available at: [Accessed 10 January 2020].

Giuliodori, M. J., 2006. Peer instruction enhanced student performance on qualitative problem solving questions. Advances in Physiology Education, Volume 30, pp.168-173.

Haapasalo, L., 2003. The conflict between conceptual and procedural knowledge: Should we need to understand in order to be able to do, or vice versa?. Bulletins of the Faculty of Education, Volume 86, pp.1-20.

Isaac-Physics, 2019. Available at: [Accessed 10 January 2020].

Konig, J., 2015. Education in Chemistry. Available at: [Accessed 10 January 2020].

Lishchynska M. et al, 2019. The Transposition Project: origins, context and early findings. MSOR Connections, 17(2), pp.8-15.

Lucariello, J., Tine, M. & Ganley, C., 2014. A formative assessment of students’ algebraic variable misconceptions. The Journal of Mathematical Behavior, Volume 33, pp.30-41.

Mazur, E., 1996. Confessions of a converted lecturer. Available at: [Accessed 10 January 2020].

Mazur, E., 1997. Peer instruction. A user's manual. Upper Saddle River: Prentice Hall.

NUMBAS, 2015. NUMBAS: really versatile maths e-assessment. Available at: [Accessed 10 January 2020].

NUMBAS-CIT-Transposition, 2019. NUMBAS Transposition Tutorial. Available at: [Accessed 10 January 2020].

O'Brien, A. and Ní Ríordáin, M., 2017. Examining difficulties in initial algebra: Pre-requisite and algebra content areas for Irish post-primary students. Dublin, 10th Congress of European Research in Mathematics Education.

O'Connor, B. and Norton, S., 2016. Investigating Students' Mathematical Difficulties with Quadratic Equations. Adelaide, 39th annual conference of the Mathematics Education Research Group of Australasia.

Pendergast M. and Treacy P., 2015. Analysis of Ireland's algebra problem. Prague, Ninth Congress of European Society for Research in Mathematics Education.

Rittle-Johnson, B., Siegler, R. S. and Alibali, M. W., 2001. Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of educational psychology, 93(2), pp 346-362.

Scott, F., 2012. Is mathematics to blame? An investigation into high school students' difficulty in performing calculations in chemistry. Chem. Educ. Res. Pract., Volume 13, pp.330-336.

Smith, M. K. et al., 2009. Why peer discussion improves student performance on in-class concept questions. Science, 323(5910), pp.122-124.

Smith, M. K., Wood, W. B., Krauter, K. and Knight, J., 2011. Combining peer discussion with instructor explanation increases student learning from in-class concept questions. CBE-Life Sciences Education, 10(1), pp.55-63.

Southall, E., 2016. The formula triangle and other problems with procedural teaching in mathematics. School Science Review, 97(360), pp.49-53.

Star, J. R., 2000. On the relationship between knowing and doing in procedural learning. Mahwah, Proceedings of fourth international conference of the Learning Sciences.

Stephens, A. C. et al., 2013. Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students’ understandings. The Journal of Mathematical Behavior, 32(2), pp.173-182.



  • There are currently no refbacks.