Skip to main content

An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks

Abstract

The multiplication principle serves as a cornerstone in enumerative combinatorics. The principle underpins many basic counting formulas and provides students with a critical element of combinatorial justification. Given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we analyze a number of university textbooks in order to explore and identify several key aspects of the various statements of the principle. We characterize the nature of the variety of statements by identifying structural, operational, and bridge statements, and we highlight the respective statements’ mathematical and pedagogical implications. We conclude by indicating several directions for future research.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    We follow a number of authors in using the term “multiplication principle” throughout the paper, even though the textbooks we surveyed had many different names for it.

  2. 2.

    She clarifies that “the word ‘concept’ (sometimes replaced by ‘notion’) will be mentioned whenever a mathematical idea is concerned in its ‘official’ form” (p. 3), whereas a “conception” refers to “the whole cluster of internal representations and associations evoked by the concept” (p. 3).

  3. 3.

    We did not include universities outside of the United States in order to limit the scope and because we did not feel equipped to linguistically analyze textbooks in other languages. We also did not examine probability textbooks, primarily because we suspect that reasoning about multiplication in probability contexts may fundamentally differ from strictly combinatorial contexts, and thus the inclusion of such texts would require further analysis and would complicate our narrative. Each of these ideas represents a potential direction for future research.

  4. 4.

    The expression 9 × 13 + 3 × 12 = 153 results from a case breakdown that attends to the issue of dependence, where the first case is if a non-heart face card is drawn first, and the second case is if a heart face card is drawn first.

  5. 5.

    A classic example of this is the relationship between the formulas for permutations and combinations, in which each selection of k objects from n distinct objects is actually equivalent to k! arrangements of k objects from n objects. To find the number of combinations we can count permutations and divide by k!

References

  1. Abrahamson, D., & Wilensky, U. (2005). Understanding chance: from student voice to learning supports in a design experiment in the domain of probability. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th annual meeting of the north American chapter of the international group for the psychology of mathematics education (Vol. 7, pp. 1–8). Roanoke: Virginia Tech University.

    Google Scholar 

  2. Annin, S. A., & Lai, K. S. (2010). Common errors in counting problems. Mathematics Teacher, 103(6), 402–409.

    Google Scholar 

  3. Ball, D. L., & Feiman-Nemser, S. (1988). Using textbooks and teachers’ guides: A dilemma for beginning teachers and teacher educators. Curriculum Inquiry, 18, 401–423.

    Article  Google Scholar 

  4. Batanero, C., Godino, J., & Navarro-Pelayo, V. (1997a). Combinatorial reasoning and its assessment. In I. Gal & J. B. Garfield (Eds.), The assessment challenge in statistics education (pp. 239–252). IOS Press.

  5. Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997b). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181–199.

    Article  Google Scholar 

  6. Begle, E. G. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66, 207–214.

    Google Scholar 

  7. Brualdi, R. A. (2010). Introductory Combinatorics (5th ed.). Upper Saddle River: Pearson Prentice Hall.

  8. Cai, J., Lo, J. J., & Watanabe, T. (2002). Intended treatment of arithmetic average in U.S. and Asian school mathematics textbooks. School Science and Mathematics, 102, 391–404.

    Article  Google Scholar 

  9. Capaldi, M. (2013). A study of abstract algebra textbooks. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 15th annual conference in undergraduate mathematics education (pp. 364–368). Portland: Portland State University.

    Google Scholar 

  10. Charalambous, C. Y., Delaney, S., Hsu, H. Y., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three different countries. Mathematical Thinking and Learning, 12, 117–151.

    Article  Google Scholar 

  11. Cook, J. P., & Stewart, S. (2014). Presentation of matrix multiplication in introductory linear algebra textbooks. In T. Fukuwa-Connelly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the 17th annual conference on research in undergraduate mathematics education (pp. 70–77). Denver: University of Northern Colorado.

    Google Scholar 

  12. Dubois, J. G. (1984). Une systematique des configurations combinatoires simples. Educational Studies in Mathematics, 15(1), 37–57.

    Article  Google Scholar 

  13. Eizenberg, M. M., & Zaslavsky, O. (2003). Cooperative problem solving in combinatorics: the inter-relations between control processes and successful solutions. The Journal of Mathematical Behavior, 22, 389–403.

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. English, L. D. (1991). Young children's combinatorics strategies. Educational Studies in Mathematics, 22, 451–474.

    Article  Google Scholar 

  16. English, L. D. (1993). Children’s strategies for solving two- and three-dimensional combinatorial problems. The Journal of Mathematical Behavior, 24(3), 255–273.

    Google Scholar 

  17. Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. ZDM, 5, 193–198.

    Google Scholar 

  18. Fischbein, E., Pampu, I., & Minzat, I. (1970). Effects of age and instruction in combinatory ability in children. British Journal of Educational Psychology, 3, 261–270.

    Article  Google Scholar 

  19. Godino, J., Batanero, C., & Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students. Educational Studies in Mathematics, 60, 3–36.

    Article  Google Scholar 

  20. Hadar, N., & Hadass, R. (1981). The road to solve combinatorial problems is strewn with pitfalls. Educational Studies in Mathematics, 12, 435–443.

    Article  Google Scholar 

  21. Halani, A. (2012). Students’ ways of thinking about enumerative combinatorics solution sets: The odometer category. In the electronic proceedings for the 15th special interest group of the MAA on research on undergraduate mathematics education (pp. 59–68) Portland, OR: Portland State University.

  22. Harel, G. (1987). Variation in linear algebra content presentations. For the Learning of Mathematics, 7(3), 29–32.

    Google Scholar 

  23. Kapur, J. N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127.

    Google Scholar 

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

  25. List of United States university campuses by enrollment. (n.d.). In Wikipedia. Retrieved 9 Dec 2014, from http://en.wikipedia.org/wiki/List_of_United_States_university_campuses_by_enrollment

  26. Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52(1), 29–55.

    Article  Google Scholar 

  27. Lockwood, E. (2011a). Student approaches to combinatorial enumeration: The role of set-oriented thinking. Unpublished doctoral dissertation, Portland State University – Oregon.

  28. Lockwood, E. (2011b). 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.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  31. Lockwood, E. (2014b). Both answers make sense! using the set of outcomes to reconcile differing answers in counting problems. Mathematics Teacher, 108(4), 296–301.

    Article  Google Scholar 

  32. 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.

    Google Scholar 

  33. 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.

    Article  Google Scholar 

  34. Lockwood, E., Swinyard, C., & 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.

    Article  Google Scholar 

  35. Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2011). Combinatorics and reasoning: representing, justifying, and building isomorphisms. New York: Springer.

    Google Scholar 

  36. 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.

    Article  Google Scholar 

  37. Math (n.d.). Retrieved 9 Dec 2014, from http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools/top-science-schools/mathematics-rankings

  38. Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: an empirical approach. Educational Studies in Mathematics, 56(2/3), 255–286.

    Article  Google Scholar 

  39. Mesa, V., & Goldstein, B. (2014). Conceptions of inverse trigonometric functions in community college lectures, textbooks, and student interviews. In T. Fukuwa-Connelly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the 17th annual conference on research in undergraduate mathematics education (pp. 885–893). Denver: University of Northern Colorado.

    Google Scholar 

  40. National Liberal Arts Colleges Rankings (n.d.). Retrieved 9 Dec 2014, from http://colleges.usnews.rankingsandreviews.com/best-colleges/rankings/national-liberal-arts-colleges

  41. National Research Council. (2004). On evaluating curricular effectiveness: judging the quality of K-12 mathematics evaluations. Washington: National Academy Press.

    Google Scholar 

  42. National Universities Rankings (n.d.). Retrieved 9 Dec 2014, from http://colleges.usnews.rankingsandreviews.com/best-colleges/rankings/national-universities

  43. Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. New York: W. W. Norton & Company, Inc.

    Google Scholar 

  44. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Two fourth-grade teachers’ use of a new mathematics text. The Elementary School Journal, 100, 331–350.

    Article  Google Scholar 

  45. Reys, B. J., Reys, R. E., & Chavez, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61(5), 61–66.

    Google Scholar 

  46. Shaughnessy, J. M. (1977). Misconceptions of probability: an experiment with a small-group, activity-based, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295–316.

    Article  Google Scholar 

  47. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

    Article  Google Scholar 

  48. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 1–41). Albany: State University of New York Press.

    Google Scholar 

  49. Strauss, A. & Corbin, J. (1998). Basics of qualitative research: techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, California: Sage Publications, Inc.

  50. Tarr, J. E., & Lannin, J. K. (2005). How can teachers build notions of conditional probability and independence? In G. A. Jones (Ed.), Exploring probability in school: challenges for teaching and learning (pp. 215–238).

  51. Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253–295.

    Article  Google Scholar 

  52. Tillema, E. S. (2011). Students’ combinatorial reasoning: the multiplication of binomials. Paper presented for the 33rd annual meeting of the international group for psychology of mathematics education in north America. Reno: University of Nevada.

    Google Scholar 

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

    Article  Google Scholar 

  54. Weinberg, A., & Wiesner, E. (2011). Understanding mathematics textbooks through reader-oriented theory. Educational Studies in Mathematics, 76(1), 49–63. doi:10.1007/s10649-010-9264-3.

    Article  Google Scholar 

Bibliography of Textbooks

  1. Allenby, R., & Slomson, A. (2011). How to count: an introduction to combinatorics (2nd ed.). Boca Raton: CRC Press.

    Google Scholar 

  2. Anderson, I. (2001a). A first course in discrete mathematics. London: Springer.

    Google Scholar 

  3. Anderson, J. (2001b). Discrete mathematics with combinatorics. Upper Saddle River: Prentice Hall.

    Google Scholar 

  4. Balakrishnan, B. K. (1995). Schaum’s outline of theory and problems of combinatorics. New York: McGraw-Hill, Inc.

    Google Scholar 

  5. Bender, E., & Williamson, S. (2006). Foundations of combinatorics with applications. Mineola: Dover Publications.

    Google Scholar 

  6. Berge, C. (1971). Principles of combinatorics. New York: Academic Press.

    Google Scholar 

  7. Berman, G., & Fryer, K. (1972). Introduction to combinatorics. New York: Academic Press, Inc.

    Google Scholar 

  8. Biggs, N. L. (1989). Discrete mathematics (2nd ed.). Oxfordshire: Clarendon Press.

    Google Scholar 

  9. Bogart, K. P. (1990). Introductory combinatorics (2nd ed.). San Diego: Harcourt Brace Jovanovich.

    Google Scholar 

  10. Bogart, K. P., Stein, C., & Drysdale, R. L. (2006). Discrete mathematics for computer science. Emeryville: Key College Publishing.

    Google Scholar 

  11. Bona, M. (2007). Introduction to enumerative combinatorics. New York: McGraw Hill.

    Google Scholar 

  12. Bose, R. C., & Manvel, B. (1984). Introduction to combinatorial theory. New York: John Wiley & Sons.

    Google Scholar 

  13. Britton, J. R., & Bello, I. (1984). Topics in contemporary mathematics (3rd ed.). New York: Harper & Row.

    Google Scholar 

  14. Brualdi, R. A. (2004). Introductory combinatorics (4th ed.). Upper Saddle River: Pearson Prentice Hall.

    Google Scholar 

  15. Cameron, P. J. (1994). Combinatorics: topics, techniques, algorithms. Cambridge: Cambridge University Press.

    Google Scholar 

  16. Chen, C., & Koh, K. (1992). Principles and techniques in combinatorics. Hackensack: World Scientific Publishing Co.

    Book  Google Scholar 

  17. Cohen, D. (1978). Basic techniques of combinatorial theory. New York: Wiley.

    Google Scholar 

  18. DeTemple, D., & Webb, W. (2014). Combinatorial reasoning: An introduction to the art of counting. Wiley.

  19. Dossey, J. A., Otto, A. D., Spence, L. E., & VandenEynden, C. (2006). Discrete mathematics (5th ed.). Boston: Pearson.

    Google Scholar 

  20. Eisen, M. (1969). Elementary combinatorial analysis. New York: Gordon and Breach.

    Google Scholar 

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

    Google Scholar 

  22. Erickson, M. (1996). Introduction to combinatorics. New York: Wiley.

    Book  Google Scholar 

  23. Ferland, K. (2009). Discrete mathematics. Boston: Houghton Mifflin Company.

    Google Scholar 

  24. Gerstein, L. J. (1987). Discrete mathematics and algebraic structures. New York: W. H. Freeman and Company.

    Google Scholar 

  25. Gersting, J. (1999). Mathematical structures for computer science (4th ed.). New York: W.H. Freeman.

    Google Scholar 

  26. Goldstein, L., & Schneider, D. (1984). Finite mathematics and its applications (2nd ed.). Englewood Cliffs: Prentice Hall.

    Google Scholar 

  27. Goodaire, E., & Parmenter, M. (1998). Discrete mathematics with graph theory. Upper Saddle River: Prentice Hall.

    Google Scholar 

  28. Gossett, E. (2002). Discrete mathematics with proof. Upper Saddle River: Prentice Hall.

    Google Scholar 

  29. Grimaldi, R. (1994). Discrete and combinatorial mathematics: an applied introduction (3rd ed.). Reading: Addison-Wesley.

    Google Scholar 

  30. Harris, J., Hirst, J. L., & Mossinghoff, M. (2008). Combinatorics and graph theory. New York: Springer.

    Book  Google Scholar 

  31. Hein, J. L. (2003). Discrete mathematics (2nd ed.). Sudbury: Jones and Bartlett Publishers.

    Google Scholar 

  32. Hillman, A. P., Alexanderson, G. L., & Grassl, R. M. (1987). Discrete and combinatorial mathematics. New Jersey: Macmillan.

    Google Scholar 

  33. Jackson, B., & Thoro, D. (1990). Applied combinatorics with problem solving. Reading: Addison-Wesley Pub.

    Google Scholar 

  34. Johnson, D. B., & Mowry, T. A. (1999). Finite mathematics: practical applications (3rd ed.). Pacific Grove: Brooks/Cole Pub.

    Google Scholar 

  35. Johnsonbaugh, R. (2001). Discrete mathematics (5th ed.). Upper Saddle River: Prentice Hall, Inc.

    Google Scholar 

  36. Koh, K. M., & Tay, E. G. (2013). Counting (2nd ed.). Hackensack: World Scientific Publishing Co.

    Book  Google Scholar 

  37. Kolman, B., Busby, R., & Ross, S. C. (2008). Discrete mathematical structures (6th ed.). Boston: Pearson.

    Google Scholar 

  38. Krussel, J. (2011). Discrete Mathematics (and Other Stuff). Unpublished document.

  39. Lial, M. L., Greenwell, R. N., & Ritchey, N. P. (2012). Finite mathematics and calculus with applications (9th ed.). Boston: Pearson.

    Google Scholar 

  40. Liu, C. L. (1968). Introduction to combinatorial mathematics. New York: McGraw Hill.

    Google Scholar 

  41. Lovasz, L., Pelikan, J., & Vesztergombi, K. (2003). Discrete mathematics: elementary and beyond. New York: Springer.

    Book  Google Scholar 

  42. Maki, D. P., & Thompson, M. (2004). Finite mathematics (4th ed.). New York: McGraw Hill.

    Google Scholar 

  43. Marcus, D. A. (1998). Combinatorics: A problem oriented approach. Washington: Springer.

    Google Scholar 

  44. Martin, G. E. (2001). The art of enumerative combinatorics. New York: Springer.

    Book  Google Scholar 

  45. Matousek, J., & Nesetril, J. (2008). Invitation to discrete mathematics (2nd ed.). New York: Oxford University Press.

    Google Scholar 

  46. Mazur, D. R. (2010). Combinatorics: a guided tour. Washington, DC: MAA.

    Google Scholar 

  47. Mott, J. L., Kandel, A., & Baker, T. P. (1986). Discrete mathematics for computer scientists and mathematicians (2nd ed.). Upper Saddle River: Prentice Hall, Inc.

    Google Scholar 

  48. Polya, G., Tarjan, R. E., & Woods, D. R. (1983). Notes on introductory combinatorics. Boston: Birkhäuser.

    Book  Google Scholar 

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

    Google Scholar 

  50. Riordan, J. (2002). An introduction to combinatorial analysis. Mineola: Dover Publications, Inc.

    Google Scholar 

  51. Roberts, F. S., & Tesman, B. (2009). Applied combinatorics (2nd ed.). Upper Saddle River: Pearson Prentice Hall.

    Google Scholar 

  52. Rolf, H., & Williams, G. (1988). Finite mathematics. Dubuque: W.C. Brown.

    Google Scholar 

  53. Rosen, K. H. (2007). Discrete mathematics and its applications (6th ed.). New York: McGraw Hill.

    Google Scholar 

  54. Ross, K., & Wright, C. (1992). Discrete mathematics (3rd ed.). Englewood Cliffs: Prentice Hall.

    Google Scholar 

  55. Ryser, H. (1963). Combinatorial mathematics. Washington: MAA.

    Google Scholar 

  56. Scheinerman, E. (2000). Mathematics: a discrete introduction. Pacific Grove: Brooks/Cole.

    Google Scholar 

  57. Slomson, A. (1991). An introduction to combinatorics. Boca Raton: Chapman & Hall/CRC.

    Google Scholar 

  58. Stanley, R. (1997). Enumerative combinatorics (Vol. I). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  59. Tan, S. T. (2008). Finite mathematics for the managerial, life, and social sciences (9th ed.). Belmont: Brooks/Cole.

    Google Scholar 

  60. Tucker, A. (2002). Applied combinatorics (4th ed.). New York: John Wiley & Sons.

    Google Scholar 

  61. Van Lint, J., & Wilson, R. M. (2001). A course in combinatorics (2nd ed.). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  62. Velleman, D. (1994). How to prove it: a structured approach. Cambridge: Cambridge University.

    Google Scholar 

  63. Vilenkin, N. Y. (1971). Combinatorics. London: Academic Press, Inc.

    Google Scholar 

  64. Wallis, W. D., & George, J. C. (2011). Introduction to combinatorics. Boca Raton: Chapman & Hall/CRC.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Elise Lockwood.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lockwood, E., Reed, Z. & Caughman, J.S. An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks. Int. J. Res. Undergrad. Math. Ed. 3, 381–416 (2017). https://doi.org/10.1007/s40753-016-0045-y

Download citation

Keywords

  • Combinatorics
  • Discrete mathematics
  • Textbook analysis
  • Reinvention