Skip to main content
SpringerLink
Log in

An Unexpected Outcome: Students’ Focus on Order in the Multiplication Principle

  • Published:
International Journal of Research in Undergraduate Mathematics Education Aims and scope Submit manuscript

Abstract

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration. In an effort to better understand students’ reasoning about the MP, we had two undergraduate students reinvent a statement of the MP in a teaching experiment. In this paper, we adopt an actor-oriented perspective (Lobato, Educational Researcher, 32(1), 17–20, 2003, Educational Psychologist, 47(3), 1–16 2012) and draw on Lockwood’s, Journal of Mathematical Behavior, 32, 251–265 (2013) model of students’ combinatorial thinking to trace the students’ development of a certain conception about order – that a statement of the MP should allow for any ordering of the stages of a counting process. Notably, this conception differs from how order is treated in textbook statements of the MP. We report on how this conception persisted for the students, and we explain how the researchers (interviewers) and students managed to proceed through the teaching experiment while maintaining different conceptions of order. The students’ reasoning about order sheds light on ways in which students may think about order and highlights potential subtleties about multiplication in counting that deserve attention.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. We do not mean to infer too much from how an expression is written in terms of actually reflecting one’s counting process. For example, someone could actually engage in the process of first considering the 5 options of the last position (so, effectively going through the process of 5⋅9⋅8⋅7⋅6) but write it as 9⋅8⋅7⋅6⋅5, and so the order in which they actually performed the stages is lost in the written inscription. Here we write 5⋅9⋅8⋅7⋅6 intentionally to describe the order of the stages, where the first stage is written first (or leftmost) and the last stage is written last (or rightmost). This is to facilitate our own communication of the stages, but we acknowledge that we cannot infer an order of stages just from a written expression without having confirmation of what the writer of that expression intends.

  2. Hour-long interviews occurred on Wednesday and Friday of Week 1, Monday and Wednesday of Week 2, Friday of Week 3, and Monday, Wednesday, and Friday of Week 4.

  3. Another solution is to say there are 64 options for the first rook and then 14 remaining places for the second rook, yielding an expression of 64⋅14.

  4. Said another way, this is a Cartesian Product problem that does not specify the order in which the elements of the k-tuple are generated or listed. That is, a (coin, card, dice) 3-tuple is an acceptable outcome, as is a (card, dice, coin) 3-tuple. Such problems involve creating k-tuples out of k predetermined, independent sets.

  5. See Lockwood et al. (2017) for more information on the statements chosen in this study and Lockwood and Purdy (2019) for more in-depth discussion of their evaluation of textbook statements.

  6. Tucker’s (2002) statement is our preferred statement because it addresses all of the elements laid out in the textbook analysis (Lockwood et al., 2017); furthermore, it is a statement that explicitly links a counting process with a set of outcomes.

References

  • Auerbach, C., & Silverstein, L. B. (2003). Qualitative data: An introduction to coding and analysis. New York, NY: New York University Press.

  • Bona, M. (2007). Introduction to enumerative Combinatorics. New York, NY: McGraw Hill.

  • Eizenberg, M. M., & Zaslavsky, O. (2004). Students’ verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15–36.

    Article  Google Scholar 

  • Epp, S. (2010). Discrete mathematics with applications, 4th edition. Boston: Brooks/Cole.

  • Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 225–273). Mahwah, NJ: Erlbaum.

  • Halani, A. (2012). Students’ ways of thinking about enumerative combinatorics solution sets: The odometer category. In (Eds.) S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education (pp. 59-68) Portland, OR: Portland State University.

  • Kavousian, S. (2008) Enquiries into undergraduate students’ understanding of combinatorial structures. Unpublished doctoral dissertation, Simon Fraser University – Vancouver, BC.

  • Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17–20.

    Article  Google Scholar 

  • Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47(3), 1–16.

    Article  Google Scholar 

  • Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical Behavior, 21, 87–116.

    Article  Google Scholar 

  • Lockwood, E. (2011). Student connections among counting problems: An exploration using actor-oriented transfer. Educational Studies in Mathematics, 78(3), 307–322. https://doi.org/10.1007/s10649-011-9320-7.

    Article  Google Scholar 

  • Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32, 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008.

    Article  Google Scholar 

  • Lockwood, E. (2014). A set-oriented perspective on solving counting problems. For the Learning of Mathematics, 34(2), 31–37.

    Google Scholar 

  • Lockwood, E., & Caughman, J. S. (2016). Set partitions and the multiplication principle. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26(2), 143–157. https://doi.org/10.1080/10511970.2015.1072118.

    Article  Google Scholar 

  • Lockwood, E., & Gibson, B. (2016). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students. Educational Studies in Mathematics, 91(2), 247–270. https://doi.org/10.1007/s10649-015-9664-5.

    Article  Google Scholar 

  • Lockwood, E., & Purdy, B. (2019). Two undergraduate students’ reinvention of the multiplication principle. Journal for Research in Mathematics Education, 3(50), 225–267.

    Article  Google Scholar 

  • Lockwood, E., Swinyard, C. A., & Caughman, J. S. (2015). Patterns, sets of outcomes, and combinatorial justification: Two students’ reinvention of counting formulas. International Journal of Research in Undergraduate Mathematics Education, 1(1), 27–62. https://doi.org/10.1007/s40753-015-0001-2.

    Article  Google Scholar 

  • Lockwood, E., Reed, Z., & Caughman, J. S. (2017). An analysis of statements of the multiplication principle in combinatorics, discrete, and finite mathematics textbooks. International Journal of Research in Undergraduate Mathematics Education, 3(3), 381–416. https://doi.org/10.1007/s40753-016-0045-y.

  • Martin, G. E. (2001). The art of enumerative Combinatorics. New York, NY: Springer.

  • Richmond, B., & Richmond, T. (2009). A discrete transition to advanced mathematics. Providence, RI: American Mathematical Society.

  • Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education. Mahwah, NJ: Lawrence Erlbaum Associates.

  • Tillema, E. S. (2013). A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems. Journal of Mathematical Behavior, 32(3), 331–352. https://doi.org/10.1016/j.jmathb.2013.03.006.

    Article  Google Scholar 

  • Tucker, A. (2002). Applied Combinatorics (4th ed.). New York, NY: Wiley.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Oregon State University, 064 Kidder Hall, Corvallis, OR, 97331, USA

    Elise Lockwood

  2. Department of Mathematics, Oregon State University, 328 Kidder Hall, Corvallis, OR, 97331, USA

    Branwen Purdy

Authors
  1. Elise Lockwood
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Branwen Purdy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elise Lockwood.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Table 4 All problems given in the teaching experiment (a similar table appears in Lockwood and Purdy (2019))
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lockwood, E., Purdy, B. An Unexpected Outcome: Students’ Focus on Order in the Multiplication Principle. Int. J. Res. Undergrad. Math. Ed. 6, 213–244 (2020). https://doi.org/10.1007/s40753-019-00107-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40753-019-00107-3

Keywords

Search

Navigation