While there is still much to be done in the area of combinatorics education, there have been a number of valuable studies conducted over the years. My colleagues and I have assembled a fairly comprehensive list of papers related to combinatorics education, available here. See the Teaching Resources page for combinatorial textbooks. If you come across any additional studies that should be on the list, or if you need help identifying any of the papers, please feel free to let me know at Elise (dot) Lockwood (at) oregonstate (dot) edu.

For those who might be wondering where to begin, below is a list of some articles that I believe could serve as a particularly good starting point.

**Research on Combinatorics Education**

- Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils.
*Educational Studies in Mathematics, 32*, 181-199. - Eizenberg, M. M., & Zaslavsky, O. (2004). Students’ verification strategies for combinatorial problems.
*Mathematical Thinking and Learning, 6*(1), 15-36. - English, L. D. (1991). Young children’s combinatorics strategies.
*Educational Studies in Mathematics, 22*, 451-47. - English, L. D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. A. Jones (Ed.),
*Exploring probability in school: Challenges for teaching and learning*(Vol. 40, pp. 121-141): Kluwer Academic Publishers. - Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2011).
*Combinatorics and reasoning: Representing, justifying, and building isomorphisms.*New York: Springer. - Mamona-Downs, J. & Downs, M. (2004). Realization of techniques in problem solving: the construction of bijections for enumeration tasks.
*Educational Studies in Mathematics, 56,*235-253. - Mellinger, K. E. (2004). Ordering elements and subsets: Examples for student understanding.
*Mathematics and Computer Education, 38*(3), 333-337. - Piaget, J., & Inhelder, B. (1975).
*The origin of the idea of chance in children*. New York: W. W. Norton & Company, Inc.

Here is a partial list of my own papers on combinatorics education. They’re organized chronologically and include journal articles and refereed conference proceedings.

**My Research on Combinatorics Education**

**Lockwood, E.**(2011). Student connections among counting problems: An exploration using actor-oriented transfer.*Educational Studies in Mathematics, 78*(3), 307-322. DOI: 10.1007/s10649-011-9320-7.**Lockwood, E.**(2012). Counting using sets of outcomes.*Mathematics Teaching in the Middle School*,*18*(3), 132-135.**Lockwood, E.**(2013). A model of students’ combinatorial thinking.*Journal of Mathematical Behavior, 32,*251-265. DOI: 1016/j.jmathb.2013.02.008.**Lockwood, E.**(2014). A set-oriented perspective on solving counting problems.*For the Learning of Mathematics, 34*(2), 31-37.**Lockwood, E.**(2014). Both answers make sense! Using the set of outcomes to reconcile differing answers in counting problems.*Mathematics Teacher*,*108*(4), 296-301.**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. Doi: 10.1007/s40753-015-0001-2.**Lockwood, E.**(2015). The strategy of solving smaller, simpler problems in the context of combinatorial enumeration.*International Journal of Research in Undergraduate Mathematics Education, 1*(3), 339-362. DOI: 10.1007/s40753-015-0016-8.**Lockwood, E.**, Swinyard, C. A., & Caughman, J. S. (2015). Modeling outcomes in combinatorial problem solving: The case of combinations. In T. Fukawa-Connelly, N. Infante, K. Keene, and M. Zandieh (Eds.),*Proceedings of the 18*(pp. 601-696). Pittsburgh, PA: West Virginia University.^{th}Annual Conference on Research on Undergraduate Mathematics Education**Lockwood, E.**& Caughman, J. S. (2016). Set partitions and the multiplication principle.*Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26*(2), 143-157. DOI: 10.1080/10511970.2015.1072118.**Lockwood, E.**, & Gibson, B. (2016). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students.*Educational Studies in Mathematics, 91*(2), 247-270. DOI: 10.1007/s10649-015-9664-5.**Lockwood, E.**& Swinyard, C. A. (2016). An introductory set of activities designed to facilitate successful combinatorial enumeration for undergraduate students.*Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26*(10), 889-904. DOI: 10.1080/10511970.2016.1194934.**Lockwood, E.**, Reed, Z., & Caughman, J. S. (2016). An analysis of statements of the multiplication principle in combinatorics, discrete, and finite mathematics textbooks. Online first,*International Journal of Research in Undergraduate Mathematics Education*. DOI: 10.1007/s40753-016-0045-y.**Lockwood, E.**, Wasserman, N., & McGuffey, W. (2016). Classifying combinations: Do students distinguish between different types of combination problems? In T. Fukawa-Connelly, N. Infante, M. Wawro, & S. Brown (Eds.),*Proceedings of the 19*(pp. 296-309)^{th}Annual Conference on Research on Undergraduate Mathematics Education,*.*Pittsburgh, PA: West Virginia University.**Lockwood, E.**& Reed, Z. (2016). Students’ meanings of a (potentially) powerful tool for generalizing in combinatorics. In T. Fukawa-Connelly, N. Infante, M. Wawro, & S. Brown (Eds.),*Proceedings of the 19*(pp. 1-16)^{th}Annual Conference on Research on Undergraduate Mathematics Education,*.*Pittsburgh, PA: West Virginia University.**Best Paper Award Winner**.**Lockwood, E.**& Schaub, B. (2016). Reinventing the multiplication principle. In T. Fukawa-Connelly, N. Infante, M. Wawro, & S. Brown (Eds.),*Proceedings of the 19*(pp. 31-45)^{th}Annual Conference on Research on Undergraduate Mathematics Education,*.*Pittsburgh, PA: West Virginia University.**Meritorious Citation.****Lockwood, E.**& Erickson, S. (2017). Undergraduate students’ initial conceptions of factorials.*International Journal of Mathematical Education in Science and Technology, 48*(4), 499-519. DOI: 10.1080/0020739X.2016.1259517.**Lockwood, E.**& Schaub, B. (2017). An unexpected outcome: Students’ focus on order in the multiplication principle. To appear in the*Proceedings of the 20*San Diego, CA: San Diego State University.^{th}Annual Conference on Research on Undergraduate Mathematics Education.**Lockwood, E.**(2017). A preliminary investigation of the reification of “choosing” in counting problems. To appear in the*Proceedings of the 20*San Diego, CA: San Diego State University.^{th}Annual Conference on Research on Undergraduate Mathematics Education.**Lockwood, E.**& Reed, Z. (In press). Reinforcing mathematical concepts and developing mathematical practices through combinatorial activity. To appear in Hart, E. W. & Sandefur, J. (Eds.)*ICME-13 Monograph on the Teaching and Learning of Discrete Mathematics.*Berlin: Springer.