Algorithms . . . Alcatraz: Are children prisoners of process?

  • Chris Hurst Curtin University


Multiplicative thinking is a critical component of mathematics which largely determines the extent to which people develop mathematical understanding beyond middle primary years. We contend that there are several major issues, one being that much teaching about multiplicative ideas is focussed on algorithms and procedures. An associated issue is the extent to which algorithms are taught without the necessary explicit connections to key mathematical ideas. This article explores the extent to which some primary students use the algorithm as a preferred choice of method and whether they can recognise and use alternative ways of calculating answers. We also consider the extent to which the students understand ideas that underpin algorithms. Our findings suggest that most students in the sample are ‘prisoners to procedures and processes’ irrespective of whether or not they understand the mathematics behind the algorithms.  


Anghileri, J. (2000). Mental and written calculation methods for multiplication and division. In M. Askew & M. Brown. (Eds.) Teaching and learning primary numeracy: Policy, practice and effectiveness. London: British Education Research Association.

Anghileri, J. (2006). Teaching number sense, 2nd Ed.. London: Continuum.

Askew, M. (2016). Transforming primary mathematics: Understanding classroom tasks, tools and talk. London: Routledge.

Australian Curriculum Assessment and Reporting Authority. (2017). Australian Curriculum: Mathematics v8.3. Retrieved from:

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27, 41-51.

Davis, B. (2008). Is 1 a prime number? Developing teacher knowledge through concept study. Mathematics Teaching in the Middle School, 14 (2), 86-91.

Department of Education (2014). Mathematics programmes of study: Key stages 1and 2. National curriculum in England. Retrieved from:

Devlin, K. (2008). It ain’t no repeated addition. Retrieved from:

Goutard, M. (2017) Inventing Division Techniques. Mathematics Teaching, 256 (42-43)

Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products of students’ own mathematical activity. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (p. 114-122). Reston, VA: NCTM.

Hartnett, J. (2015). Teaching computations in primary school without traditional written algorithms. In M. Marshman, V. Gieger, & A. Bennison (Eds.). Mathematics education in the margins. (Proceedings of the 38th annual conference of the Mathematics Education Research group of Australasia), pp. 285-292. Sunshine Coast: MERGA.

Jazby, D., & Pearn, C. (2015). Using alternative multiplication algorithms to offload cognition. In M. Marshman, V. Geiger, & A. Bennison (Eds.). Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 309-316. Sunshine Coast: MERGA

Kamii, C. (1989). Young Children Continue to Reinvent Arithmetic, Second Grade: Implications of Piaget’s Theory. New York: Teachers College Press.
National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center). (2010). Common core state standards for mathematics. Retrieved from:

Pearn, C. (2009). Highlighting the similarities and differences of the mathematical knowledge and strategies of Year 4 students. In R. Hunter, B. Bicknell, & T. Burgess (Eds) Crossing divides (Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia, pp. 443-451) Wellington: MERGA.

Pesek, D. D. & Kirshner, D., (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31, 524-540.

Ross, S. (2002). Place value: Problem solving and written assessment. Teaching Children Mathematics, March, 2002, 419-423.

Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding Numeracy in the Middle Years – Project Findings, Materials, and Resources, Final Report submitted to Victorian Department of Education and Training and the Tasmanian Department of Education, Retrieved from

Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching 77 (20-26).

Swan, P. (2004). Computational choices made by children: What, why and how. Australian Primary Mathematics Classroom, 9(3), 27-30.

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th Ed.). Boston: Pearson.

Warren, E., & English, L. (2000). Primary school children's knowledge of arithmetic structure. In J. Bana & A. Chapman (Eds.), Mathematics Education beyond 2000 (Proceedings of 23rd annual conference of the Mathematics Education Research Group of Australasia, Fremantle, pp. 624-631). Sydney: MERGA.

Young-Loveridge, J., & Mills, J. (2009). Teaching Multi-digit Multiplication using Array-based Materials. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 2). Palmerston North, NZ: MERGA.